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Chapter 11 – Permutations, Combinations, and the Binomial Theorem ● 1 Pre-Calculus 12 11.1 Permutations The Fundamental Counting Principle If one item can be selected in m ways, and for each way a second item can be selected in n ways, then the two items can be selected in ____________ ways. Example 1: You are packing clothing to go on a trip, however the airline has restricted how much you can carry-on . And so you are only able to fit 3 different tops, 2 pair of pants 2 pairs of shoes in your baggage. Determine the number of different outfits consisting of a top, a pair of pants and a pair of shoes are possible? Example 2: How many even natural numbers are there with 3 digits? Example 3: How many license plates can be made with 6 characters in which the first 2 of the characters are letters and the rest are digits? Example 4: A multiple-choice test has 7 questions, with 4 possible answers for each question. Suppose students answer each question by guessing randomly. What is the probability that all 7 questions will be answered correctly?

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Page 1: Chapter 11 Permutations, Combinations and the Binomial Theoremmathlau.weebly.com/uploads/1/4/9/9/14997226/chapter_11... · 2018-10-15 · Chapter 11 – Permutations, Combinations,

Chapter 11 – Permutations, Combinations, and the Binomial Theorem ● 1

Pre-Calculus 12

11.1 Permutations

The Fundamental Counting Principle If one item can be selected in m ways, and for each way a second item can be selected in n ways, then the two items can be selected in ____________ ways.

Example 1: You are packing clothing to go on a trip, however the airline has restricted how much you can carry-on . And so you are only able to fit 3 different tops, 2 pair of pants 2 pairs of shoes in your baggage. Determine the number of different outfits consisting of a top, a pair of pants and a pair of shoes are possible?

Example 2: How many even natural numbers are there with 3 digits? Example 3: How many license plates can be made with 6 characters in which the first 2 of the characters are letters and the rest are digits?

Example 4: A multiple-choice test has 7 questions, with 4 possible answers for each question. Suppose students answer each question by guessing randomly. What is the probability that all 7 questions will be answered correctly?

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2 ● Chapter 11 – Permutations, Combinations, and the Binomial Theorem

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Example 5: How many ways can you arrange 5 different books on a shelf? Factorial Notation the ___________of consecutive natural numbers in decreasing order to the number one The symbol ! is used in mathematics to denote the factorial operation. 1! = 2! = 3! = 4! = 5! = n! = Definition of 0! 0! = Why?

Example 5: Without a calculator, evaluate !4!58

!60

Permutations (Rule A) – with distinct objects • An _________________ arrangement of ______________________ is called a permutation.

• The number of permutations of n distinct objects is .

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

Notation: =25 P

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Example 6: From a group of 100 people, how many ways can a president, vice-president, and treasurer be selected? Example 7: Solve each equation for n. c) 422 =Pn d) 9010 =nP

Permutations (Rule B) - with Identical Objects The number of permutations of n objects of which there are a objects alike of one kind, b alike of another kind, c alike of another kind, and so on, is

Example 8: Determine the number of permutations of all the letters in the word, STATISTICIAN. Example 9: On the following grid, how many different paths can A take to get to B, assuming one can only travel east and south? Explain.

A

B

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Permutations with Constraints Example 10: Chris, Sandy, Kurtis, Theresa and Andrea go to watch a movie and sit in 5 adjacent seats. In how many ways can this be done under each condition? a) without restriction b) all the boys and all the girls sit together

c) if Chris sits next to Sandy d) if Sandy refuses to sit next to Chris Example 11: How many 4 digit odd number can you make using the digits 1 to 7 if the numbers must be less than 6000? No digits are repeated. Assignment: page 524-527 #1-28

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Chapter 11 – Permutations, Combinations, and the Binomial Theorem ● 5

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11.2 Combinations

Combinations • An unordered arrangement of distinct objects is called a combination.

• The number of combinations of n distinct objects taken r at a time is .

Example 1: a) How many different committees of 3 people can be formed from 7 people? b) How many different committees of 3 people can be formed from 7 people if the first person selected

serves as the chairperson, the second as the treasurer, and the third as the secretary? c) If the group of 7 people consists of 3 males and 4 females, how many different committees of 3 people

can be formed with 1 male and 2 females? Think: you must choose 1 male out of the group of 3 males and 2 females out of the group of 4 females.

d) If the group of 7 people consists of 3 males and 4 females, how many different committees of 3 people

can be formed with at least one male on the committee?

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Pre-Calculus 12

Example 2: Solve for n a) 452 =Cn b) 2811 =−+ nn C

Example 3: A standard deck of 52 playing cards consists of 4 suits (spaces, hearts, diamonds, and clubs) of 13 cards each. a) How many different 5-card hands can be formed? b) How many different 5-card hands can be formed that consist of all hearts? c) How many different 5-card hands can be formed that consist of all face cards? d) How many different 5-card hands can be formed that consist of 3 hearts and 2 spades?

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e) How many different 5-card hands can be formed that consist of exactly 3 hearts?

f) How many different 5-card hands can be formed that consist of at least 3 hearts? Example 4: A basketball coach has five guards and seven forwards on his basketball team. a) In how many different ways can he select a starting line-up of two guards and three forwards? b) How many starting teams are there if the star player, who plays guard, must be included? Example 6: During a basketball tournament, all players shake hands with each other at the end of the last game. If 300 handshakes were exchanged, how many were at the tournament?

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Mixing it up! a) If there are 8 horses in a race, how many

different ways can 3 horses advance to the next round?

b) If there are 8 horses in a race, how many different ways can 3 horses be awarded 1st, 2nd, and 3rd place?

c) Calculate the number of ways a president, vice president and a treasurer be selected from a class of 30 students.

d) Calculate the number of ways a group of 3 people be selected for student council from a class of 30 students.

e) How many 3-letter words are there of the word SAMPLE?

f) How many ways are there for all the letters in PARALLEL?

g) A multiple choice test has 8 questions, with 4 possible answers for each question. Suppose the answer to each question is a guess. How many different ways are there to complete the test?

h) A multiple choice test has 8 questions, with 3 answers are A, 2 answers are B, 2 answers are C, and 1 answer is D. How many different answer keys are possible?

Assignment: page 534-536 #1-2m 3(cd), 4-11, 13-15, 17-21

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Pascal’s Triangle

Pascal’s Triangle is full of patterns however did you noticed each of the numbers can be written using combinations?

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11.3 The Binomial Theorem

Investigate the Patterns in Binomial Powers: a) Expand and simplify each of the following powers of the binomial a + b:

( ) =+ 2ba

( ) =+ 3ba

( ) =+ 4ba b) What pattern do you see in the

• powers of a?

• powers of b?

• numerical coefficients in the expansions?

c) Expand and simplify: ( ) =+ 5ba

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Pre-Calculus 12

Visualizing – The Binomial Expansion of (a + b)4

( ) 4044

3134

2224

1314

0404

4 baCbaCbaCbaCbaCba ++++=+

0 b’s 1 b 2 b’s 3 b’s 4 b’s Number of ways to choose this many b’s from 4 factors of (a + b) Example 1: Expand

a) ( ) =+ 41x

b) ( ) =− 412x The Binomial Theorem (using combinations)

For any whole number n:

(a + b)n = nC0 anb0 + nC1 a

n-1b1 + nC2 a n-2b2 + … + nCk a

n-kbk + … + nCn a0bn

1st term 2nd term 3rd term term (n+1)th term For this expansion the (k + 1)th term is: =+1kt .

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Example 2: Determine the 4th term in the expansion of ( )93−x .

Example 3: Write the first four terms of the binomial expansion of ( )122yx + .

Example 4: Find the term in x3 in ( )1221 x− .

Example 5: One term in the expansion of ( )8ax + is 448x6. Determine the value of a.

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Example 6: The 4th term in the expression of n

x

−2

1 is 715x− . Determine n.

Constant Term

If we expand 3

2 1

+a

a we get,

( ) ( ) ( ) ( )

( ) ( )

3

36

32

246

302

33

212

23

122

13

032

03

32

133

11

13

131

11111

aaa

aaa

aaa

aaC

aaC

aaC

aaC

aa

+++

+

+

+=

+

+

+

=

+

Example 7: Without expanding the binomial out, determine the constant term in the expansion of

62 1

−x

x .

Assignment: page 542-543 #1-2(ac), 3(b), 4-7, 11-12, 14, 17-20

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14 ● Chapter 11 – Permutations, Combinations, and the Binomial Theorem

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