Permutations and Combinations - Wikispacesmyresearchunderwood.wikispaces.com/file/view/ch13.pdfChapter 13 Permutations and Combinations 717 INTRODUCTION THIS CHAPTER IS AN introduction

TERMINOLOGY

13 Permutations and Combinations

Arrangements: Different ways of organising objects

Combinations: Arrangements of objects without replacement or repetition when order is not important. The notation used is Cn r for selecting r objects from n where order doesn’t matter

Factorial: A factorial is the product of n consecutive positive integers from n down to one. For example 6! = 6 x 5 x 4 x 3 x 2 x 1

Fundamental counting principle: If one event can occur in p ways and a second independent event can occur in q ways, then the two successive events can occur in p x q different ways

Ordered selections: Selections that are taken in a particular position

Permutations: The arrangement of objects without replacement or repetition when order is important. The notation used is Pr

n for selecting r objects from n where order matters

Random experiments: Experiments that are made with no pattern or order where each outcome is equally likely to occur

Sample space: The set of all possible outcomes in an event or series of events

Unordered selections: Selections that are made when the order of arrangements is not important or relevant

ch13.indd 716 7/11/09 12:21:35 AM

717Chapter 13 Permutations and Combinations

INTRODUCTION

THIS CHAPTER IS AN introduction to some of the concepts you will meet in probability in the HSC Course. Probability is the study of how likely it is that something will happen. It is used to make predictions in different areas, ranging from games of chance to business decision-making.

In this chapter you will study general counting techniques based on the fundamental counting principle. These will lead on to the study of permutations and combinations. These have applications in fi nding the size of the sample space, or the ways that certain events can happen in probability. It can also tell us the number of postcodes a state can have, the number of telephone numbers that is possible in a city and the number of combinations making up serial numbers for appliances.

The probability of an event E happening, P ( E ), is given by the number of ways the event can occur, n ( E ), compared with the total number of outcomes possible n ( S ) (the size of the sample space).

( )( )

EPn Sn E

=] g

If 0P E =] g the event is impossible. If 1P E =] g the event is certain (it has to happen). P E0 1# #] g

Fundamental Counting Principle

Simple probability

You have studied probability in earlier stages of mathematics. We can measure probability in theory. However, probability only gives us an approximate idea of the likelihood of certain events happening.

For example, in Lotto draws, there is a machine that draws out the balls at random and a panel of supervisors checks that this happens properly. Each ball is independent of the others and is equally likely to be drawn out.

In a horse race, it is diffi cult to measure probability as the horses are not all equally likely to win. Other factors such as ability, training, experience and weight of the jockey all affect it. The likelihood of any one horse winning is not random.

ch13.indd 717 7/11/09 12:21:40 AM

718 Maths In Focus Mathematics Extension 1 Preliminary Course

EXAMPLES

1. Alison buys 5 raffl e tickets and 100 are sold altogether. What is the probability that Alison (a) wins (b) doesn’t win fi rst prize in the raffl e?

Solution

The size of the sample space, or total number of outcomes is 100, (a) since there are 100 tickets altogether. Alison has 5 tickets so has 5 different ways of winning the raffl e.

( )P

1005

201

Win =

=

There are (b) 100 5- or 95 other tickets that could win if Alison loses.

( )P

10095

2019

Loss =

=

Or, if we know that the sum of all probabilities is 1, we could say

( ) ( )P P1

1201

2019

Loss Win= -

= -

=

2. There are 56 books on music at the school library and there are 2000 books altogether. If Anthony selects a book at random, fi nd the probability that it will be a book on music.

Solution

The size of the sample space is 2000 and there are 56 ways that Anthony could select a music book.

( )P

200056

2507

Music book =

=

The sum of all probabilities is 1. Complementary events: –P E P E1not =] ]g g or ( )P E P E1= -^ hL where EL is the complement of E P E P E 1+ =] ^g hL

ch13.indd 718 7/12/09 12:24:55 AM


1. A lottery is held in which 20 000 tickets are sold. If I buy 2 tickets, what is the probability of my winning the prize in the lottery?

2. The probability of a bus arriving

on time is estimated at 3317 . What

is the probability that the bus will not arrive on time?

3. The probability of a seed

producing a pink fl ower is 97 .

Find the probability that the seed will produce a different coloured fl ower.

4. In a lottery, 200 000 tickets are sold. If Lucia buys 10 tickets, what is the probability of her winning fi rst prize?

5. A machine has a 1.5% chance of breaking down at any given time. What is the probability of the machine not breaking down?

6. A bag contains 6 red balls and 8 white balls. If one ball is drawn out at random, fi nd the probability that it will be

white (a) red. (b)

7. A shoe shop orders in 20 pairs of black, 14 pairs of navy and 3 pairs of brown school shoes. If the boxes are all mixed up, fi nd the probability that one box selected at random will contain brown shoes.

8. A biased coin is weighted so that heads comes up twice as often as tails. Find the probability of tossing a tail.

9. A die has the centre dot painted white on the 5 so that it appears as a 4. Find the probability of rolling

a 2 (a) a 4 (b) a number less than 5. (c)

10. A book has 124 pages. If the book is opened at any page at random, fi nd the probability of the page number being

either 80 or 90 (a) a (b) multiple of 10 an odd number (c) less than 100. (d)

11. In the game of pool, there are 15 balls, each with the number 1 to 15 on it. In Kelly Pool, each person chooses a number at

random from a container and has to try and sink the ball with the corresponding number. If Tracey chooses a number, fi nd the probability that her ball will be

the eight ball (a) an odd number (b) a number less than 10. (c)

13.1 Exercises

A multiple of 10 is a number that is divisible by 10.

ch13.indd 719 7/11/09 12:21:45 AM


12. A box containing a light globe

has a 201 probability of holding

a defective globe. If 160 boxes are checked, how many globes would be expected to be defective?

13. There are 29 red, 17 blue, 21 yellow and 19 green jelly beans in a packet. If Kate chooses one at random, fi nd the probability that it will be

red (a) blue or yellow (b) not green. (c)

14. The probability of breeding a

white budgie is 152 . If Mr Seed

breeds 240 budgies over the year, how many would be expected to be white?

15. A die is rolled. Calculate the probability of rolling

a 6 (a) an even number (b) a number less than 3 (c) 4 or more (d) a multiple of 2. (e)

16. The probability that an arrow will

hit a target is 1813 .

Find the probability that it (a) will miss the target.

If 126 arrows are fi red, how (b) many would be expected to hit a target?

17. A dog can catch a ball in its mouth 8 times out of 10.

What is the probability of the (a) dog catching a ball?

If the ball is thrown to the (b) dog 45 times, how many times would the dog be expected to catch it?

18. In a bag there are 21 lollies with pink wrappers and 23 with blue wrappers. If Leila chooses a lolly at random from the bag, fi nd the probability that she selects a lolly with

a blue wrapper (a) a pink wrapper. (b)

19. In a survey, it was found that 18 people preferred Brand A of soft drink while 39 people preferred Brand B. What is the probability that a person chosen at random prefers Brand A?

20. A school has 653 junior and 329 senior students. If a student is chosen at random, what is the probability that it will be a senior student?

21. A class has 12 girls and 19 boys. Eight of the girls and 12 of the boys play a sport. If a student is chosen at random, fi nd the probability that the person chosen

is a boy (a) plays a sport (b) is a girl who doesn’t play (c)

a sport doesn’t play a sport. (d)

22. Amie’s CD collection includes 21 R&B, 14 rock and 24 jazz albums. If she selects one CD to play at random, fi nd the probability that the CD is

rock (a) R&B or rock (b) jazz (c) not R&B. (d)

ch13.indd 720 7/11/09 12:21:46 AM


Counting techniques

In the HSC Course you will learn about multi-stage events (events where there are two or more events such as rolling two dice or tossing two coins). The counting can become quite diffi cult, which is why we introduce counting techniques in the Preliminary Course.

23. The probability of winning a

game of chance is 83 and the

probability of losing is 125 . What

is the probability of a draw?

24. In a poll, 39% said they would vote Labor, 34% said they would vote Liberal and 8% said they would vote for independents or small parties.

What percentage would vote for Greens?

25. An arrow has a 0.37 probability of hitting a target outside the bullseye zone and 0.12 probability of missing the target altogether. What is the probability of the arrow hitting the bullseye area?

Class Discussion

Break up into pairs and try these experiments with one doing the activity and one recording the results.

Toss two coins as many times as you can in a 5-minute period and 1. record the results in the table:

Result Two heads One head and one tail Two tails

Tally

Compare your results with others in the class. What do you notice? Is this surprising?

Roll two dice as many times as you can in a 5-minute period, fi nd 2. the total of the two uppermost numbers on the dice and record the results in the table:

Total 2 3 4 5 6 7 8 9 10 11 12

Tally

Compare your results with others in the class. What do you notice? Is this surprising?

Why don’t these results appear to be equally likely?

ch13.indd 721 7/11/09 12:21:47 AM


There are many examples of where counting techniques are useful, in probability and in areas such as manufacturing, business, biology and economics. For example, in genetics, the number of molecules on DNA strands can be diffi cult to fi nd.

Investigation

To travel to work, Cassie needs to catch a bus and a train. She lives 1. near a bus stop and there are three different buses she could catch into town. When she arrives in town, she needs to catch one of four trains to work.

If there are three buses and four trains possible for Cassie to catch, in how many ways is it possible for her to travel to work?

Buses Trains

A 1

C

B234

Cassie’shouse

At a restaurant, there are three entrees, four main meals and two 2. desserts. Every time Rick eats at the restaurant he chooses to eat a different combination of courses. How many times would he need to go to the restaurant to cover all possible combinations?

FUNDAMENTAL COUNTING PRINCIPLE

If one event can happen in a different ways, a second event can happen in b different ways, a third in c different ways and so on, then these successive events can happen in abc … different ways.

EXAMPLES

1. A personal identifi cation number (PIN) has 4 letters followed by 3 numbers. How many different PINs of this type are possible?

Solution

There are 26 letters and 10 numbers ( –0 9 ) possible for the positions in the PIN.

ch13.indd 722 7/11/09 12:21:47 AM


Here are some examples of counting when there is no repetition or replacement.

EXAMPLES

1. To win a trifecta in a race, a person has to pick the horses that come fi rst, second and third in the race. If a race has 9 horses, how many different combinations could be a trifecta?

Solution

Any of the 9 horses could come fi rst. Any of the remaining 8 could come second. Any of the remaining 7 horses could come third.

9 8 7

504

Total ways # #=

=

26 26 26 26 10 10 10

26 10456 976 000

Total number4 3

# # # # # #

#

=

=

=

So 456 976 000 PINs are possible.

2. A restaurant serves 5 different types of entree, 12 main courses and 6 desserts.

If I order any combination of entree, main course and dessert at (a) random, how many different combinations are possible?

If my friend makes 3 guesses at which combination I will order, what (b) is the probability that she will guess correctly?

Solution

(a) 5 12 6

360

Total number of combinations =

=

# #

(b) P360

3

1201

correct guess =

=

^ h

The probabilities will be different for where each horse will come

in the race, but the number of possible

different trifecta combinations will be

the same.

CONTINUED

ch13.indd 723 7/11/09 12:21:47 AM


2. A group of 15 people attend a concert and 3 of them are randomly given a free backstage pass. The fi rst person receives a gold pass, the second one a silver pass and the third one a bronze pass. In how many different ways can the passes be given out?

Solution

Any of the 15 people can receive the fi rst pass. There are 14 people left who could receive the second pass. Similarly there are 13 people that could receive the third pass.

15 14 13

2 730

Total number of possibilities # #=

=

3. In Lotto, a machine contains 45 balls, each with a different number from 1 to 45.

In how many ways can 6 balls be randomly drawn? (a) To win fi rst prize in Lotto, a person must choose all 6 numbers (b)

correctly. Lisa has 3 tickets in the same draw of Lotto. What is the probability that she will win fi rst prize?

Solution

The fi rst ball could be any of the 45 balls. (a) The second could be any of the remaining 44 balls and so on.

45 44 43 42 41 40

5 864 443 200

The number of ways # # # # #=

=

(b) P5 864 443 200

3

1 954 814 4001

first prize =

=

^ h

1. A password has 4 letters. How many combinations are possible?

2. A motorcycle numberplate is made up of 2 letters followed by 2 numbers. How many numberplates of this type are available?

3. A password can have up to 5 letters followed by 4 numbers on it. If I could use any letter of the alphabet or number, how many different passwords could be formed? Leave your answer in index form.

13.2 Exercises

ch13.indd 724 7/11/09 12:21:48 AM


4. A witness saw most of the numberplate on a getaway car except for the fi rst letter and the last number. How many different cars do the police need to check in order to fi nd this car?

5. A certain brand of computer has a serial number made up of 10 letters then 15 numbers. How many computers with this type of serial number can be made? Leave your answer in index form.

6. Victoria has postcodes starting with 3. How many different postcodes are available in Victoria?

7. A country town has telephone numbers starting with 63 followed by any 6 other numbers from 0 to 9. How many telephone numbers are possible in this town?

8. Yasmin has 12 tops, 5 pairs of jeans and 5 pairs of shoes in her wardrobe. If she randomly chooses a top, pair of jeans and shoes, how many combinations are possible?

9. A car manufacturer produces cars in 8 different colours, with either manual or automatic gear transmission, and 4 different types of wheels. How many different combinations can it produce?

10. A PIN has 4 numbers. If I forget my PIN I am allowed 3 tries to get it right. Find the probability that I get it within the 3 tries .

11. A restaurant offers 7 main courses and 4 desserts, as well as 3 different types of coffee.

How many different (a) combinations of main

course, dessert and coffee are possible?

Find the probability that (b) I randomly pick the combination voted most favourite.

12. A telephone number in a capital city can start with a 9 and has 8 digits altogether.

How many telephone (a) numbers are possible?

If I forget the last 3 digits of (b) my friend’s telephone number, how many numbers would I have to try for the correct number?

13. A company manufactures 20 000 000 computer chips. If it uses a serial number on each one consisting of 10 letters, will there be enough combinations for all these chips?

14. A password consists of 2 letters followed by 5 numbers. What is the probability that I randomly guess the correct password?

15. A city has a population of 3 500 000. How many digits should its telephone numbers have so that every person can have one?

16. A manufacturer of computer parts puts a serial number on

each part, consisting of 3 letters, 4 numbers then 4 letters. The number of parts sold is estimated as 5 million. Will there be enough combinations on this serial number to cope with these sales?

17. A bridal shop carries 12 different types of bridal dresses, 18 types of veils and 24 different types of shoes. If Kate chooses a combination of dress, veil and

ch13.indd 725 7/11/09 12:21:48 AM


Factorial notation

Counting outcomes when repetition or replacement is allowed is quite straightforward, even when the numbers become very large.

shoes at random, what is the probability that she chooses the same combination as her friend Jane?

18. Kate chooses a different coloured dress for each of her 3 bridesmaids. If the colours are randomly given to each bridesmaid, how many different possibilities are there?

19. In a computer car race game, the cars that come fi rst, second and third are randomly awarded. If there are 20 cars, how many possible combinations of fi rst, second and third are there?

20. Jacquie only has 4 chocolates left and decides to randomly choose which of her 6 friends will receive one each. How many possible ways are there in which can she give the chocolates away?

21. Three prizes are given away at a concert by taping them underneath random seats. If there are 200 people in the audience, in how many ways can these prizes be won?

22. There are 7 clients at a hairdressing salon. If there are

3 free haircuts randomly given away, in how many ways could this be done?

23. A fl ock of 28 pelicans is fed 6 fi sh carcasses. If each carcass is given to a different pelican, in how many ways can this happen?

24. A set of cards is numbered 1 to 100 and 2 chosen at random.

How many different (a) arrangements of ordered pairs are possible?

What is the probability that (b) a particular ordered pair is chosen?

25. Each of 10 cards has a letter written on it from A to J. If 3 cards are selected in order at random, fi nd the probability that they spell out CAB.

EXAMPLE

A card is drawn randomly from a set of 25 cards numbered 1 to 25 in turn and then replaced before the next is selected. How many possible outcomes are there if 25 cards are chosen this way? Answer in scientifi c notation, correct to 3 signifi cant fi gures.

ch13.indd 726 7/25/09 12:09:32 PM


When there is no repetition or replacement, the calculations can be quite long.

Solution

Each time there is a card drawn, there are 25 possibilities.

. . .

.

25 25 25 25 25

25

8 88 10

Total number times25

34

# # # #

#

=

=

=

] g

EXAMPLE

A card is drawn randomly from a set of 25 cards numbered 1 to 25 in turn without replacing it before the next is drawn. How many possible outcomes are there if all 25 cards are drawn out? Answer in scientifi c notation, correct to 3 signifi cant fi gures.

Solution

First card: there are 25 possibilities. Second card: there are only 24 possibilities since one card has already been drawn out . Third card: there are 23 possibilities and so on.

…

.

25 24 23 3 2 1

1 55 10

Total number25

# # # # # #

#

=

=

Check 0! on your calculator.

Factorial notation allows us to easily calculate the number of possible outcomes when selecting all objects in order with no replacement or repetition.

! . . .n n n n n n1 2 3 4 3 2 1# #= - - - -] ] ] ]g g g g

Since the sequence of numbers multiplied doesn’t go further than 1, then by convention we say that

0! 1=

This calculation is quite tedious!

You can fi nd a x!

key on most scientifi c calculators.

ch13.indd 727 7/11/09 12:21:50 AM


EXAMPLES

1. Evaluate 4! (a) 7! (b) 25! (answer in scientifi c notation correct to 3 signifi cant fi gures.) (c)

Solution

(a) !4 4 3 2 124# # #=

=

(b) !7 7 6 5 4 3 2 15040# # # # # #=

=

(c) ! .25 1 55 1025#=

2. A group of 9 teenagers is waiting to be served in a café. They are each randomly assigned a number from 1 to 9.

In how many ways is it possible for the numbers to be assigned? (a) One of the group needs to be served quickly as he has to leave. (b)

If he is given the fi rst number, in how many ways is it possible for the numbers to be assigned?

Solution

The fi rst number could be assigned 9 ways. (a) The second number could be assigned 8 ways and so on.

!9

362 880Total ways =

=

One of the group is given the fi rst ticket (this can only happen in one (b) way) The second number could be assigned 8 ways and so on.

!1 8

40 320Total ways #=

=

It is much easier to use the

x! key on a calculator to

fi nd this.

1. Evaluate 6! (a) 10! (b) 0! (c) (d) ! !8 7- (e) !5 4#

(f) 4!7!

(g) 5!12!

(h) 4!9!13!

(i) 3!5!8!

(j) 4!7!11!

13.3 Exercises

ch13.indd 728 7/11/09 12:21:51 AM


2. A group of 9 jockeys are each given a set of riding colours to wear. If these are given out in order randomly, how many different arrangements are possible?

3. Each of 6 people at a restaurant is given a different coloured glass. How many possible combinations are there?

4. A mountain trail only has room for one person at a time. If 12 people are waiting at the bottom of the trail and are randomly picked to start out, in how many ways can this happen?

5. A dog walker has 5 dogs and 5 leashes. In how many different ways is it possible to put a leash on each dog?

6. There are 11 people in a choir and each receives a musical score.

In how many different ways (a) could the scores be handed out?

Russell, the musical director, (b) also needs a musical score. In how many ways could the scores be handed out for the choir and the musical director?

7. A row of seats in a theatre seats 8 people. In how many ways could a group of 8 friends be randomly seated in this row?

8. A group of 7 people line up to do karaoke. If they are each randomly given a song to sing, how many possible outcomes are there?

9. A kindergarten class has a rabbit, a mouse and a parrot. Three children are selected to take these pets home for the holidays. If the pets are randomly given out to these children, how many different ways are possible?

10. A PIN consists of 6 numbers, with no repetition of any numbers allowed. How many different PINs are possible?

11. In a chorus for a school musical, 7 students each wear a different mask. In how many different ways can the masks be worn by these students?

12. If 15 people play a game of Kelly pool, each person in turn chooses a number at random between 1 and 15. In how many different ways can this occur? Answer in scientifi c notation, correct to one decimal place.

13. (a) A school talent quest has 11 performers and each one

is randomly given the order in which to perform. In how many ways can the order of performances be arranged?

If one performer is chosen to (b) perform fi rst, in how many ways can the others be arranged?

14. A group of 6 friends sit in the same row at a concert.

In how many different ways (a) can they arrange themselves?

If one friend must sit on the (b) aisle, in how many ways can they be arranged?

ch13.indd 729 7/11/09 12:21:51 AM


15. A group of 8 friends go to a restaurant and sit at a round table. If the fi rst person can sit anywhere, in how many ways can the others be arranged around the table?

16. In a pack of cards, the 4 aces are taken out and shuffl ed.

What is the probability of (a) picking out the Ace of Hearts at random?

If all the aces are arranged in (b) order, what is the probability of guessing the correct order?

17. At a wedding, each of the 12 tables is to have a centrepiece with a different coloured rose.

In how many different ways (a) can the roses be arranged at random?

What is the probability that (b) the bride will have a pink rose at her table?

18. In a maths exam, a student has to arrange 5 decimals in the correct order. If he has no idea how to do this and arranges them randomly, what is the probability that he makes the right guess for all the decimals?

19. In a car race, the fastest car is given pole position and the other cars are randomly given their starting positions. If there are 14 cars altogether, in how many ways can this be arranged?

20. Show that

(a) !!

48 8 7 6 5# # #=

(b) !!

611 11 10 9 8 7# # # #=

(c) !!

rn

n n n n1 2 3= - - -] ] ]g g g... r n r1 where 2+] g

(d) ( ) !

!n r

n-

n n n n1 2 3= - - -] ] ]g g g... n r n r1 where 2- +] g

Permutations

Factorial notation is useful for fi nding the number of possible outcomes when arranging all objects in order without replacement. However, sometimes we need to fi nd the number of possible outcomes when arranging only some of the objects in order without replacement.

It is easy to arrange objects with replacement.

ch13.indd 730 7/11/09 12:21:52 AM


EXAMPLE

In how many ways can 5 cards be selected from a total of 20 cards if each one is replaced before selecting the next one?

Solution

Each selection can be made in 20 possible ways.

.

20 20 20 20 20

20

3 200 000

Total

ways

5

=

=

=

# # # #

For r selections from n objects (with repetitions), the number of possible outcomes is n n n n# # # f ( r times) or n r

However, when arranging r objects from n objects in order without replacement, it is not so easy.

EXAMPLE

In how many ways can 5 cards be selected from a total of 20 cards if there is no replacement?

Solution

The 1 st card can be selected in 20 different ways. The 2 nd card can be selected in 19 different ways as the fi rst card is no longer being used. The 3 rd card can be selected in 18 different ways, and so on.

20 19 18 17 16

1 860 480

Total

ways

=

=

# # # #

The calculations can become tedious if we select a larger number of objects.

ch13.indd 731 7/11/09 12:21:54 AM


EXAMPLE

If there are 20 cards and 13 cards are chosen in order at random without replacement, fi nd the possible number of ways the cards can be chosen in scientifi c notation correct to 1 decimal place.

Solution

The fi rst card can be any of the 20 numbers. The second card can be any of the remaining 19 numbers. The third can be any of the remaining 18 numbers.

20 19 18 17 8

4.8 10

The number of ways the cards can be chosen14

=

=

# # # # #

#

f

For r ordered selections from n objects without replacement, the number of possible outcomes is 1 2 3n n n n r times- - -# # # f] ] ] ]g g g g or 1n n n n n r1 2 3- - - - +f] ] ] ]g g g g

Permutation Pnr is the number of ways of making ordered selections

of r objects from a total of n objects.

!

!Pn r

nnr = -] g

A permutation describes an arrangement of r objects from a total of n objects in a certain order without replacement or repetition.

Proof

!!

P n n n n n r

n n n n n rn r n r n r

n r n r n r

n r n r n r

n n n n n r n r n r n r

n r

n

1 2 3 1

1 2 3 11 2 3 2 1

1 2 3 2 1

1 2 3 2 1

1 2 3 1 1 2 3 2 1

nr = - - - - +

= - - - - +- - - - -

- - - - -

=- - - - -

- - - - + - - - - -

=-

#$ $

$ $

$ $

$ $

f

ff

f

f

f f

] ] ] ]] ] ] ] ] ] ]

] ] ]

] ] ]] ] ] ] ] ] ]

]

g g g gg g g g g g g

g g g

g g gg g g g g g g

g

A special case of this result is:

You can fi nd a Pn

r key on most scientifi c calculators.

!P nnn =

ch13.indd 732 7/11/09 12:21:54 AM


Proof

!!

!!

!!

!

!

Pn rn

Pn nn

n

n

n

0

1

n

nn

r = -

=-

=

=

=

`

]

]

g

g

EXAMPLES

1. Evaluate P94

Solution

!!

!!

P9 4

9

59

5 4 3 2 19 8 7 6 5 4 3 2 1

9 8 7 6

3024

94

$ $ $ $

$ $ $ $ $ $ $ $

$ $ $

=-

=

=

=

=

] g

2. (a) Find the number of arrangements of 3 digits that can be formed using the digits 0 to 9 if each digit can only be used once.

(b) How many 3 digit numbers greater than 700 can be formed?

Solution

There are 10 digits from 0 to 9. (a) The 1 st digit can be any of the 10 digits. The 2 nd digit can be any of the remaining 9 digits. The 3 rd digit can be any of the remaining 8 digits.

10 9 8

720

Total permutations =

=

# #

!!

!!

P10 3

10

710

720

or 103 = -

=

=

] g

CONTINUED

You can evaluate this on a calculator.

ch13.indd 733 7/25/09 12:09:35 PM


The 1 st digit must be 7 or 8 or 9 (3 possible digits). (b) The 2 nd digit can be any of the remaining 9 digits. The 3 rd digit can be any of the remaining 8 digits.

3 9 8

216

Total arrangements =

=

# #

Another method: There are 3 ways to get the 1 st digit. The possible arrangements of the remaining 2 digits is P9

2

33 72

216

PTotal arrangements 92=

=

=

#

#

There are some special examples that need very careful counting, such as arrangements around a circle. Others involve counting when there are identical objects.

EXAMPLES

1. (a) In how many ways can 6 people sit around a circular table? (b) If seating is random, fi nd the probability that 3 particular people

will sit together.

Solution

The 1 st person can sit anywhere around the table so we only need to (a) arrange the other 5 people.

The 2 nd person can sit in any of the 5 remaining seats. The 3 rd person can sit in any of the remaining 4 seats and so on.

5!

120


=

ch13.indd 734 7/11/09 12:21:55 AM


The 3 people can sit anywhere around the table together in (b) 3 2 1# # or 3! ways. The remaining 3 people can sit together in 3! ways.

3! 3!

36

Total arrangements #=

=

( )

.

P 312036

103

sit together =

=

2. In how many ways can the letters of the word EXCEPTIONAL be arranged?

Solution

EXCEPTIONAL has 11 letters with the letter E repeated. If each E was different, i.e. E 1 and E 2 , then there would be 11! arrangements. However, we cannot tell the difference between the 2 Es. Since there are 2! ways of arranging the Es, then there are 2! arrangements of the word EXCEPTIONAL that are identical. We need to divide by 2! to eliminate these identical arrangements.

!!

.2

11

19 958 400


=

The number of different ways of arranging n objects in which a of the objects are of one kind, b objects are of another

kind, c of another kind and so on, is given by ! ! !

!a b c

nf

where

a b c nf#+ + +

EXAMPLE

Find the number of ways that the word ANAETHEMA can be arranged.

Solution

There are 9 letters, including 3 As and 2 Es. There are 9! ways of arranging the letters, with 3! ways of arranging the As and 2! ways of arranging the Es.

! !!

3 29

30 240


=

ch13.indd 735 7/11/09 12:21:55 AM


Some questions involving counting need different approaches and sometimes it is just a matter of logically working it out.

EXAMPLES

A bag contains 5 balls of different colours—red, yellow, blue, green and white. In how many ways can these 5 balls be arranged

with no restrictions (a) if the yellow ball must be fi rst (b) if the fi rst ball must not be red or white (c) if blue and green must be together (d) if red, blue and green must be together? (e)

Solution

The 1 st can be any of the 5 balls. (a) The 2 nd can be any of the remaining 4 balls and so on.

5!120

Total arrangements ==

The 1 st ball must be yellow, so there is only 1 way of arranging this. (b) The 2 nd ball can be any of the remaining 4 balls. The 3 rd ball can be any of the remaining 3 balls and so on.

4!24

Total arrangements ==

The 1(c) st ball could be yellow, blue or green so there are 3 possible arrangements. The 2 nd ball could be any of the remaining 4 balls and so on.

3 4!72

Total arrangements ==#

When two objects must be together, we treat them as a single object (d) with 2! possible arrangements. So we arrange 4 balls in 4! ways: R, Y, BG and W. But there are 2! ways in which to arrange the blue and green balls.

4! 2!48


When three objects are together, we treat them as a single object with (e) 3! possible arrangements. We are then arranging 3 balls in 3! ways: RBG, Y, W. But there are 3! ways in which to arrange the red, blue and green balls.

3! 3!36


ch13.indd 736 7/11/09 12:21:56 AM


1. Write each permutation in factorial notation and then evaluate.

(a) P63

(b) P52

(c) P83

(d) P107

(e) P96

(f) P75

(g) P86

(h) P118

(i) P91

(j) P66

2. A set of 26 cards, each with a different letter of the alphabet, is placed into a hat and cards drawn out at random. Find the number of ‘words’ possible if selecting

2 cards (a) 3 cards (b) 4 cards (c) 5 cards. (d)

3. A 3 digit number is randomly made from cards containing the numbers 0 to 9.

In how many ways can this (a) be done if the cards cannot be used more than once and zero cannot be the fi rst number?

How many numbers over (b) 400 can be made?

How many numbers less than (c) 300 can be made?

4. A set of 5 cards, each with a number from 1 to 5 on it, is placed in a box and 2 drawn out at random. Find the possible number of combinations

altogether (a) of numbers greater than (b)

50 possible

of odd numbers (c) of even numbers. (d)

5. (a) How many arrangements of the letters A, B, C and D are possible if no letter can be used twice?

(b) How many arrangements of any 3 of these letters are possible?

6. A 4 digit number is to be selected at random from the numbers 0 to 9 with no repetition of digits.

How many arrangements can (a) there be?

How many arrangements of (b) numbers over 6000 are there?

How many arrangements (c) of numbers less than 8000 are there?

7. The numbers 1, 2, 3, 4 and 5 are arranged in a line. How many arrangements are possible if

there is no restriction (a) the number is less than (b)

30 000 the number is greater than (c)

20 000 the number is odd (d) any 3 numbers are selected at (e)

random?

8. There are 12 swimmers in a race. In how many ways could they (a)

fi nish? In how many ways could they (b)

come in fi rst, second and third?

9. How many different ordered arrangements can be made from the word COMPUTER with

2 letters (a) 3 letters (b) 4 letters? (c)

13.4 Exercises

The fi rst number cannot be zero.

The fi rst number is not zero.

ch13.indd 737 7/11/09 12:30:43 AM


10. How many different ordered arrangements can be made from these words?

CENTIPEDE (a) ALGEBRA (b) TELEVISION (c) ANTARCTICA (d) DONOR (e) BASKETBALL (f) GREEDY (g) DUTIFUL (h) MANUFACTURER (i) AEROPLANE (j)

11. A group of friends queue outside a restaurant in a straight line. Find how many ways the friends can be arranged if there are

4 friends (a) 7 friends (b) 8 friends (c) 10 friends (d) 11 friends. (e)

12. A group of friends go into a restaurant and are seated around a circular table. Find how many arrangements are possible if there are

4 friends (a) 7 friends (b) 8 friends (c) 10 friends (d) 11 friends. (e)

13. A string of beads looks the same if turned over. Find the number of different arrangements possible with

10 beads (a) 12 beads (b) 9 beads (c) 11 beads (d) 13 beads. (e)

14. In how many ways can a group of 6 people be arranged

in a line (a) in a circle? (b)

15. Find how many different ways a group of 9 people can be arranged in

a line (a) a circle. (b)

16. In how many ways can a set of 10 beads be arranged

in a line (a) in a circle around the edge of (b)

a poster on a bracelet? (c)

17. (a) How many different arrangements can be made from the playing cards Jack, Queen, King and Ace?

(b) If I choose 2 of these cards randomly, how many different arrangements could I make?

(c) If I choose 3 of these cards randomly, how many different arrangements could I make?

18. A group of 7 people sit around a table. In how many ways can they be arranged

with no restrictions (a) if 2 people want to sit (b)

together if 2 people cannot sit (c)

together if 3 people sit together? (d)

19. A group of 5 boys and 5 girls line up outside a cinema. In how many ways can they be arranged

with no restriction? (a) (b) If a particular girl stands in

line fi rst? (c) If they alternate between boys

and girls (with either a girl or boy in fi rst place)?

20. Find the probability that if 10 people sit around a table, 2 particular people will be seated together at random.

All beads are different from each other.

ch13.indd 738 7/11/09 12:21:56 AM


Vowels are letters a, e, i, o and u while consonants are all other letters.

21. A bookshelf is to hold 5 mathematics books, 8 novels and 7 cookbooks.

In how many different ways (a) could they be arranged? (Leave answer in factorial notation.)

If the books are grouped in (b) categories, in how many ways can they be arranged? (Answer in factorial notation.)

If one book is chosen at (c) random, fi nd the probability that it is a cookbook.

22. (a) How many different arrangements can be made from the numbers 3, 4, 4, 5 and 6?

(b) How many arrangements form numbers greater than 4000?

(c) How many form numbers less than 5000?

(d) If an arrangement is made at random, fi nd the probability that it is less than 4000.

23. Find the probability that an arrangement of the word LAPTOP will start with T.

24. What is the probability that if a 3 letter ‘word’ is formed randomly from the letters of PHYSICAL, it will be CAL?

25. A minbus has 6 forward facing and 2 backward facing seats. If 8 people use the bus, in how many ways can they be seated

with no restrictions (a) if one person must sit in a (b)

forward facing seat if 2 people must sit in a (c)

forward facing seat?

26. If 3 letters of the word VALUED are selected at random, fi nd the number of possible arrangements if

the fi rst letter is D. (a) the fi rst letter is a vowel. (b)

27. The letters of the word THEORY are arranged randomly. Find the number of arrangements.

with no restrictions. (a) if the E is at the beginning. (b) if the fi rst letter is a (c)

consonant and the last letter is a vowel.

28. Find the number of arrangements possible if x people are

in a straight line (a) in a circle (b) in a circle with 2 people (c)

together in a straight line with (d)

3 people together in a circle with 2 people not (e)

together.

29. (a) Use factorial notation to

show that ! !

P P

3 5

83

85

=

(b) Prove that ! !r

P

n r

Pnr

nn r

=-

-

] g

30. Prove that P P r Pnr

nr

nr

11= ++

-

ch13.indd 739 7/11/09 12:21:57 AM


EXAMPLES

1. A committee of 2 is chosen from Scott, Rachel and Kate. In how many ways can this be done?

Solution

P

6Number of ordered arrangements 3

2=

=

However, a committee of Scott and Rachel is the same as a committee of Rachel and Scott. This is the same for all other arrangements of the committee. There are 2! ways of arranging each committee of two people. To get the number of unordered arrangements, we divide the number of ordered arrangements by 2!

!

P

23

Total arrangements3

2=

=

2. There are 3 vacancies on a school council and 8 people who are available. If the vacancies are fi lled randomly, in how many ways can this happen?

Solution

PNumber of orderedarrangements 83=

However, order is not necessary here, since the 3 vacancies fi lled by, say, Hamish, Amie and Marcus, would be the same in any order. There are 3! different ways of arranging Hamish, Amie and Marcus.

!

P

356

So total arrangements8

3=

=

The number of ways of making unordered selections of r

objects from n is !r

Pnr which is the same as

! !!

n r rn-] g

Combinations

The permutation Pnr is the number of arrangements possible for an ordered

selection of r objects from a total of n objects. When the order is not important, for example when AB is the same as BA ,

the number of arrangements is called a combination .

ch13.indd 740 7/11/09 12:21:58 AM


Proof

Pn r is the ordered selection of r objects from n objects. There are r ! ways of arranging r objects. If order is unimportant, the unordered selection of r objects from n is given

by !

.r

Pn r

! !!

!

!!

!

! !!

r

P

rn rn

n rn

r

n r rn

1

nr=

-

=-

=-

#

]

]

]

g

g

g

Combination Cn r or nra k is the number of ways of making

unordered selections of r objects from a total of n objects.

! !

!Cn r r

nnr = -] g

We can call this ‘choose’ notation.

EXAMPLES

1. A bag contains 3 white and 2 black counters labelled W 1 , W 2 , W 3 and B 1 , B 2 . If two counters are drawn out of the bag, in how many ways can this happen if order is not important?

Solution

Possible arrangements (unordered) are:

W B

W B

W W W W W B B B

W W W B

W B

W B1 1

1 2

1 2 2 3 3 1 1 2

1 3 2 1 3 2

2 2

There are 10 different combinations. Using combinations, the number of different arrangements of choosing 2 counters from 5 is .C5

2

!2!(5 2)5!

3!2!5!

10

C52 =

=

=

-

CONTINUED

ch13.indd 741 7/11/09 11:13:27 PM


2. If 12 coins are tossed, fi nd the number of ways of tossing 7 tails.

Solution

The order is not important. There are C12

7 ways of tossing 7 tails from 12 coins

!7!(12 7)12!

5!7!12!

792

C127 =

=

=

-

3. (a) A committee of 5 people is formed randomly from a group of 15 students. In how many different ways can the committee be formed? (b) If the group consists of 9 senior and 6 junior students, in how many ways can the committee be formed if it is to have 3 senior and 2 junior students in it?

Solution

The order of the committee is not important. (a)

3003

15Number of arrangements =

=

5b l

3 senior students can be chosen in (b) 93b l or 84 ways.

2 junior students can be chosen in 62b l or 15 ways.

93

62

84 15

1260

Total number of arrangements =

=

=

#

#

c cm m

4. A team of 6 men and 5 women is chosen at random from a group of 10 men and 9 women. If Kaye and Greg both hope to be chosen in the team, fi nd the probability that

both will be chosen (a) neither will be chosen. (b)

Solution

C C

210 126

26 460

The number of possible teams 106

95#

#

=

=

=

For Kaye to be chosen, then 4 out of the other 8 women will be chosen (a) i.e. C8

4 For Greg to be chosen, 5 out of the other 9 men will be chosen i.e. C9

5

ch13.indd 742 7/11/09 11:13:29 PM


8820

C C70 126

26 4608820

31

Number of combinations

Probability

84

95=

=

=

=

=

#

#

For Kaye and Greg not to be included, then 5 out of the other (b) 8 women and 6 out of the other 9 men will be chosen.

C C56 84

4704

26 4604704

458

Number of combinations

Probability

85

96=

=

=

=

=

#

#

1. Write in factorial notation and evaluate.

(a) 95b l

(b) 127b l

(c) 83b l

(d) C104

(e) C115

2. (a) Evaluate (i) C10

0

(ii) C70

(iii) 140b l

(iv) C99

(v) 1111c m

(b) Hence complete (i) Cn 0 =

(ii) Cn n =

3. Find the number of different ways that a committee of 6

people can be made randomly from a group of

8 people (a) 9 people (b) 11 people (c) 15 people (d) 20 people. (e)

4. (a) A set of 3 red cards and 3 blue cards are placed in a box. By naming the red cards R 1 , R 2 and R 3 and the blue cards B 1 , B 2 and B 3 , list the number of different arrangements possible when 2 cards are drawn out at random,

with order not important. How many arrangements are possible?

(b) If there are 10 red and 10 blue cards and 7 are drawn out at random, how many different combinations are possible?

5. A coin is tossed 20 times. How many different arrangements are there for tossing 5 heads?

13.5 Exercises

The cards are all distinct from

each other.

ch13.indd 743 7/13/09 11:01:39 AM


6. A set of 10 marbles are placed in a bag and 6 selected at random. In how many different ways can this happen?

7. In poker, 5 cards are dealt from a pack of 52 playing cards. How many different arrangements are possible?

8. Three cards are drawn randomly from a set of 10 cards with the numbers 0 to 9 on them. How many different arrangements are possible if order is

important (a) unimportant? (b)

9. A debating team of 3 is chosen from a class of 14 students. In how many ways can the team be selected if order is

important (a) unimportant? (b)

10. A bag contains 12 different lollies with blue wrappers and 15 different lollies with red wrappers. If I take 6 lollies out of the bag, how many different combinations are possible?

11. A team of 4 players is chosen at random from a group of 20 tennis players to play an exhibition match. In how many ways could the team be chosen?

12. A group of 3 students to go on a student representative council is chosen at random from a class of 27. In how many different ways could this be done?

13. A board of 8 people is chosen from a membership of 35. How many different combinations are possible?

14. A basketball team of 5 players is selected at random from a group of 12 PE students.

In how many ways can the (a) team be selected?

Find the probability that Erik (b) is selected as one of the team members.

Find the probability that Erik (c) and Jens are both selected.

15. A committee of 6 people is to be selected randomly from a group of 11 men and 12 women. Find the number of possible committees if

there is no restriction on who (a) is on the committee

all committee members are to (b) be male

all members are to be female (c) there are to be 3 men and (d)

3 women a particular woman is (e)

included a particular man is not (f)

included there are to be 4 women and (g)

2 men.

16. A horse race has 15 horses competing and at the TAB, a quinella pays out on the horses that come in fi rst and second, in either order. Ryan decides to bet

on all possible combinations of quinellas. If it costs him $1 a bet, how much does he pay?

17. A group of 25 students consist of 11 who play a musical instrument and 14 who don’t. Find the number of different arrangements possible if a group of 9 students is selected at random

with no restriction (a) who all play musical (b)

instruments where 5 play musical (c)

instruments where 2 don’t play musical (d)

instruments.

ch13.indd 744 7/11/09 12:22:00 AM


18. A set of cards consists of 8 yellow and 7 red cards.

If 10 cards are selected at (a) random, fi nd the number of different arrangements possible.

If 8 cards are selected, fi nd (b) the number of arrangements of selecting

4 yellow cards (i) 6 yellow cards (ii) 7 yellow cards (iii) 5 red cards. (iv)

19. Ten cards are selected randomly from a set of 52 playing cards. Find the number of combinations selected if

there are no restrictions (a) (answer in scientifi c notation correct to 3 signifi cant fi gures)

they are all hearts (b) there are 7 hearts (c) they are all red cards (d) there are 4 aces. (e)

20. An animal refuge has 17 dogs and 21 cats. If a nursing home orders 12 animals at random, fi nd the number of ways that the order would have

7 dogs (a) 9 dogs (b) 10 dogs (c) 4 cats (d) 6 cats. (e)

21. There are 8 white, 9 red and 5 blue marbles in a bag and 7 are drawn out at random. Find the number of arrangements possible

with no restriction (a) if all marbles are red (b) if there are 3 white and 2 red (c)

marbles if there are 4 red and 1 blue (d)

marbles if there are 4 white and 2 blue (e)

marbles.

22. Out of a group of 25 students, 7 walk to school, 12 catch a train and 6 catch a bus. If 6 students are selected, fi nd the number of combinations if

all walk to school (a) none catch a bus (b) 3 walk to school and (c)

1 catches a bus 1 walks to school and 4 catch (d)

a train 3 catch a train and 1 catches (e)

a bus.

23. At a karaoke night, a group of 14 friends decide that 4 of them will sing a song together. Of the friends, 5 have previously sung this song before. In how many ways can they do this if they select

friends who have all sung the (a) song previously

2 of the friends who sang the (b) song previously

none of the friends who sang (c) the song previously?

24. (a) Evaluate C125

Evaluate (b) C127

By using factorial notation, (c) show why C C12

512

7=

25. By evaluating both sides, show that C C C9

68

68

5= +

26. Show that 13 136=7a ak k

27. Show that 10 94

93= +4b b bl l l

28. Prove that n n r=-rnb bl l

29. Prove that C!P rnr

nr=

30. Prove that n nk

n11

1=

--

+-

k kb b bl l l

All marbles are distinct.

The cards are all distinct from each

other.

ch13.indd 745 7/11/09 12:22:00 AM


1. Find the number of ways of arranging 7 people

in a straight line (a) in a circle. (b)

2. A bag contains 8 red, 5 green and 9 yellow marbles. If a marble is chosen at random, fi nd the probability that it is

red (a) green or red (b) not green. (c)

3. A carriage has 2 seats facing forwards and 2 seats facing backwards. Find the number of ways of seating 4 people in the carriage if

there are no restrictions (a) one person must sit facing forwards. (b)

4. A set of 10 cards, numbered 1 to 10, is placed into a box and 3 drawn out at random. Find the number of arrangements possible if order is

important (a) unimportant. (b)

5. A group of 10 boys and 16 girls are on a school excursion. Five of them are chosen at random to help the teacher check the rolls. Find the number of ways these students could be selected if

there is no restriction (a) 3 are girls (b) all are boys. (c)

6. The probability of winning a game of chance is 59% and the probability of a draw is 12%. What is the probability of losing?

7. In how many ways can 3 letter ‘words’ be selected at random from the word RANDOM?

8. A set of cards numbered from 1 to 20 is arranged randomly. In how many ways can this be done? (Answer in scientifi c notation, to 2 signifi cant fi gures.)

9. The probability of a missile hitting a

target is 98 . What is the probability of the

missile missing the target?

10. A 5 person committee is selected from a class of 30 students. In how many ways can the committee be selected?

11. In a horse race, a person bets at the TAB on a trifecta. To win, the person must pick the fi rst 3 horses in order. In how many ways is it possible to win if there are 11 horses in the race?

12. (a) A group of 9 friends go out to dinner and sit at a round table. In how many ways can this be done?

(b) The 9 friends then go to a nightclub and randomly queue up to get in. How many arrangements are there?

(c) If Jack and Jill queue up together, in how many ways can the friends line up?

(d) Once they get into the nightclub, there is only a table for 3 available, and the others will have to stand up. If the friends randomly assign who sits at the table, in how many different ways can they be seated?

13. How many different arrangements are there of the word

PERMUTATION (a) COMBINATION (b) FACTORIAL (c) PROBABILITY (d) SELECTION (e)

Test yourself 13

ch13.indd 746 7/11/09 12:22:00 AM


14. A set of n coins are tossed. Find the number of ways if tossing k tails.

15. A set of 20 cards is numbered 1 to 20 and 6 selected at random. Find the number of arrangements of selecting

all odd numbers (a) the last 2 numbers less than 5. (b)

16. In Australian Idyll, there are 12 singers who must choose a song to sing from a list of 32 songs. Each singer takes turns in order to randomly choose a song. In how many ways could these choices be made? Answer in scientifi c notation correct to 2 decimal places.

17. A ballet class has 30 students in it. Of these students, 21 are practising for a

ballet exam. If 8 students are chosen at random, fi nd the number of ways that

5 are practising for the exam (a) all are practising for the exam (b) 3 are practising for the exam . (c)

18. Evaluate

(a) 64a k

(b) P97

19. A serial number is made up of 4 letters and 2 numbers. If zero is not allowed, fi nd how many serial numbers are possible.

20. (a) Evaluate 0!

(b) Show that nn

= n0a ak k

1. Numbers are formed from the digits 1, 2, 3, 3, and 7 at random.

In how many ways can they be (a) arranged with no restrictions?

In how many ways can they be (b) arranged to form a number greater than 30 000?

2. A charm bracelet has 6 charms on it. In how many ways can the charms be arranged if the bracelet

has a clasp (a) has no clasp? (b)

3. Show that n n 1

1nk

1=

--

+-

k kb b bl l l for

.k n1 1-# #

4. A group of n people sit around a circular table.

In how many ways can they be (a) arranged?

How many arrangements are possible (b) if k people sit together?

5. (a) How many different arrangements of the word CHALLENGE are there?

(b) How many different arrangements are possible if 3 letters are randomly selected from the word CHALLENGE and arranged into ‘words’?

6. A subcommittee of 5 people is formed from the 12 members of a board.

If this is a random selection, in how (a) many different ways can the committee be formed?

If there are 4 NSW members and (b) 3 Queensland members on the board, what is the probability that 2 NSW and 2 Queensland members will be on the committee?

Challenge Exercise 13

ch13.indd 747 7/11/09 12:22:01 AM


7. Prove that Cn!P rnr r=

8. A management committee is made up of 5 athletes and 3 managers. If the committee is formed randomly from a group of 20 athletes and 10 managers, fi nd

the number of different ways in (a) which the committee could be formed

the probability that Marcus, an (b) athlete, is included

the probability that both Marcus and (c) his girlfriend, Rachel who is a manager, are included

the probability that Marcus and (d) Rachel are excluded from the committee.

9. A set of 100 counters, numbered from 1 to 100, is placed in a bag and 4 drawn

out at random in order. Find the number of different possible arrangements if

there is no restriction (a) all the numbers are 90 or more (b) all numbers are even (c) all numbers are less than 20 (d) the fi rst number is greater than 60 (e) the fi rst 2 numbers are odd. (f)

10. In a group of 35 students, 18 play soccer and 21 play basketball. All students play at least one of these sports. If one of these students is selected at random, fi nd the probability that this student

plays both soccer and basketball (a) plays basketball but not soccer. (b)

ch13.indd 748 7/11/09 11:13:29 PM

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Permutations and Combinations - Wikispacesmyresearchunderwood.wikispaces.com/file/view/ch13.pdfChapter 13 Permutations and Combinations 717 INTRODUCTION THIS CHAPTER IS AN introduction