13-1 Permutations and Combinations. The branch of mathematics that studies different possibilities...

13-1 Permutations and Combinations

The branch of mathematics that studies different possibilities for the arrangement of objects is

COMBINATORICS

A) 82

B) 42

C) 62

D) 22

B

A) 3

B) 55

C) 45

D) 56 56

Assume that Andrew is scheduling courses for his new semester. He has a choice of 3 history course, 2 math courses and 3 science courses. If he has to choose one course from each area, how many ways can he build his schedule?

Since the selection of one course does not affect the choice of a different course, these events are INDEPENDENT EVENTS.

The classic way to determine the number of possible ways he can build his schedule would be to make a tree.

History 1

History 2

History 3

Math 2

Math 1

Math 1

Math 2

Math 1

Math 2

Science 1Science 2Science 3

Science 1Science 2Science 3

Science 1Science 2Science 3Science 1Science 2Science 3

Science 1Science 2Science 3

Science 1Science 2Science 3

To make a yogurt parfait, you choose one flavor of yogurt, one fruit topping, and one nut topping. How many parfait choices are there?

Yogurt Parfait

(choose 1 of each)

FlavorPlain

Vanilla

FruitPeaches

Strawberries

Bananas

Raspberries

Blueberries

NutsAlmonds

Peanuts

Walnuts

The Fundamental (or Basic) Counting Principle

If a task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice, r selections for the third choice, and the choice for the first selection does not affect the others, then the task of making these selections can be done in

different ways.

If a license plate consists of a letter, then 5 numbers, how many different types of license plates are possible?

license plates

If a license plate consists of a letter, then 5 numbers. The letter cannot be an E or an F and the last number must cannot be zero. How many different types of license plates are possible?

24 10 10 10 10 9

2160000

A “make-your-own-adventure” story lets you choose 6 starting points, gives 4 plot choices, and then has 5 possible endings. How many adventures are there?

There are 120 adventures.

A permutation is an arrangement of objects in a certain order. Order is VERY important.

The number of permutations possible depends on the number of objects available (n) and the number that will be used (r).

Imagine you have 8 children’s books, each with a different title.

How many different ways can they be arranged on a shelf?

Consider the number of choices for each slot.

8 7 6 5 4 3 2 1

Multiply the number of choices for each slot, and that is the number of possible permutations, that is , the number of possible different arrangements.

So if you have n distinct items and you plan to use all of them, you can find the number of possible arrangements ( permutations) by the following multiplication.

n · (n – 1) · (n – 2) · (n – 3) · ... · 1.

This expression is called n factorial, and is written as n!.

!55(5-1)(5-2)(5-3)(5-4)

5(4)(3)(2)(1) 120

How many different arrangements are possible if 4 students are to be seated in a single row of 4 chairs?

4 3 2 1

!4

If Chloe has 7 different stuffed toys and wants to arrange them in a line on her bed, but only 3 will fit, how many different arrangments are possible.

Different problem, How?

One way to find possible permutations is to use the Fundamental Counting Principle.

First Toy

Second Toy

Third Toy

There are 7 toys. You are choosing 3 of them in order.

7 choices

6 choices

5 choices =

210 permutations

arrangements of 4 4! 4 · 3 · 2 · 1

Another way to find the possible permutations is to use factorials. You can divide the total number of arrangements by the number of arrangements that are not used. In the previous slide, there are 7 total toys and 4 whose arrangements do not matter.

arrangements of 7 = 7! = 7 · 6 · 5 · 4 · 3 · 2 · 1 = 210

This can be generalized as a formula, which is useful for large numbers of items.

The number of permutations of n distinct objects, taken r at a time, is

rn P

Number of Permutations of n Distinct ObjectsThe number of different arrangements from selecting r objects from a set of n objects (r < n), in which

1. the n objects are distinct

2. once an object is used, it cannot be repeated3. order is important

is given by the formula

!!

rn

nPrn

The number of permutations of n objects taken n at a time

Surprising?

!!

nn

nPnn

!nPnn

Evaluate: 310 P

There are 1320 permutations.

312 P

How many ways can a student government select a president, vice president, secretary, and treasurer from a group of 6 people?

This is the equivalent of selecting and arranging 4 items from 6.

= 6 • 5 • 4 • 3 = 360

There are 360 ways to select the 4 people.

Awards are given out at a costume party. How many ways can “most creative,” “silliest,” and “best” costume be awarded to 8 contestants if no one gets more than one award?

= 8 • 7 • 6

= 336

There are 336 ways to arrange the awards.

How many ways can a 2-digit number be formed by using only the digits 5–9 and by each digit being used only once?

= 5 • 4

= 20

There are 20 ways for the numbers to be formed.

Permutations in a circleIf n distinct objects are arranged around a circle, then there are

(n-1)! circular permutations of the objects.

EX:1. How many different ways can 12 students be placed around a circular table?

(12-1)! = 11!

2. In how many different ways can 7 appetizers be arranged around a circular tray?

(7-1)! = 6!

Exception to the circular permutation rule:

1) If the circle contains some sort of a fixed point, the arrangement is treated as a linear permutation.

Permutations with repetitions

If a group of objects contains a number of identical objects, then the number of permutations possible can be found by the following

Where a,b, etc. are the number of times each repeating object repeats.

!

! !.... .

n

a b etc

A combination is a grouping of items in which order does not matter. There are generally fewer ways to select items when order does not matter.

6 permutations {ABC, ACB, BAC, BCA, CAB, CBA}

1 combination {ABC}

For example, consider the 3 letters A,B, and C.,

There are 6 ways to order 3 items, so there are 6 permutations.

But they are all the same combination:

To find the number of combinations, the formula for permutations can be modified.

Because order does not matter, divide the number of permutations by the number of ways to arrange the selected items.

Because in a combination, different arrangements of the same items are not counted

The Number of Combinations of n Distinct Objects Taken r at a TimeThe number of different arrangements from selecting r objects from a set of n objects (r < n), in which

1. the n objects are distinct

2. once an object is used, it cannot be repeated

3. order is not important

is given by the formula

)!(!

!

rnr

nCrn

Evaluate: 310C

How many committees of 3 people can be formed from out of 8 people?

1. the 8 people are distinct2. once a person is chosen, he/she cannot be chosen again3. order is not importantCombinations of 8 taken 3 at a time:

When deciding whether to use permutations or combinations, first decide whether order is important. Use a permutation if order matters and a combination if order does not matter.

There are 12 different-colored cubes in a bag. How many ways can Randall draw a set of 4 cubes from the bag?

Step 1 Determine whether the problem represents a permutation of combination.

The order does not matter. The cubes may be drawn in any order. It is a combination.

= 495

There are 495 ways to draw 4 cubes from 12.

5

Step 2 Use the formula for combinations.

The swim team has 8 swimmers. Two swimmers will be selected to swim in the first heat. How many ways can the swimmers be selected?

Step 1 Determine whether the problem represents a permutation of combination.

The order does not matter. The cubes may be drawn in any order. It is a combination.

The swimmers can be selected in 28 ways.

1. the 8 persons are distinct2. once a person is chosen, he/she cannot be chosen again

3. order IS IMPORTANT

How many committees of 3 people (chair, secretary, treasurer) can be formed out of 8 people?

Permutations of 8 taken 3 at a time:

1. Six different books will be displayed in the library window. How many different arrangements are there?

2. The code for a lock consists of 5 digits. The last number cannot be 0 or 1. How many different codes are possible? 80,000

720

3. The three best essays in a contest will receive gold, silver, and bronze stars. There are 10 essays. In how many ways can the prizes be awarded?

4. In a talent show, the top 3 performers of 15 will advance to the next round. In how many ways can this be done? 455

720