Grade 12 Pre-Calculus Mathematics

[MPC40S]

Chapter 11

Permutations, Combinations and the

Binomial Theorem

Outcomes

P1, P2, P3, P4

12P.P.1. Apply the fundamental counting principle to solve problems. 12P.P.2 Determine the number of permutations of 𝑛 elements taken 𝑟 at a time to solve problems. 12.P.P.3. Determine the number of combinations of 𝑛 different elements take 𝑟 at a time to solve problems. 12P.P.4 Expand powers of a binomial in a variety of ways, including using the binomial theorem (restricted to exponents that are natural numbers).

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Chapter 11 – Homework

Section Page Questions

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Chapter 11: PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM

11.1 – Permutations (Part 1) Example #1

Suppose you are at a restaurant and you can choose one of three main entrees – Pizza, Hamburger or Sushi and one of three desserts – Ice Cream, Cake or Fruit. How many possible combo meals of one entrée and one dessert can you order?

• We could make a complete list of all the possible meal combinations…

__________ & _________ __________ & _________ __________ & __________ __________ & _________ __________ & _________ __________ & __________ __________ & _________ __________ & _________ __________ & __________

• What if there were 25 options for entrees and 13 options for desserts? What if there were also salad options? We need a better method than making a list.

P1 P2 P3

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Fundamental Counting Principle: If one task can be performed in _______________ and another task in _____________,

then both tasks can be performed in ___________________.

Example #2

How many outfits can you make if you have 4 choices of pants, 5 choices of shirts and 6 choices of shoes? Example #3

A store needs to hire three cashiers. How many ways can the store manager hire for these three positions given there are twelve candidates? Example #4

A standard license plate in Manitoba is three letters followed by three numbers. How many license plates are possible in Manitoba under these conditions?

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Example #5

How many 3-digit numbers can you make using the digits 1, 2, 3, 4 and 5 if: a) Repetition is allowed b) Repetition is not allowed

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Example #6

You bought five house numbers from Home Depot. The numbers are 1, 2, 3, 4 and 5. a) How many three-digit house numbers can you form using these numbers? b) How many three-digit house numbers can you form, if the house number must be even? c) How many three-digit house numbers can you form, if the house number must be odd? d) How many three-digit house numbers can you form, if the house number must be greater than 300? e) How many three-digit house numbers can you form, if the house number must be even and greater than 300?

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Factorial Notation: _____________________________________________________ ______________________________________________________________________ ______________________________________________________________________ Example #7

Evaluate each of the following. a) 4! = b) 8!

Example #8

There are 6 friends who must be arranged in a row. How many different ways can this be done? Example #9

Simplify each of the following.

a) 5!

3! b)

11!

9! c)

5!

7!

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Example #10

Simplify each of the following.

a) 𝑛!

(𝑛−2)! b)

(𝑛+1)!

𝑛!

Example #10

Solve for n.

a) 𝑛!

(𝑛−2)!= 30 b)

(𝑛+1)!

(𝑛−1)!− 42 = 0

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Chapter 11: PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM

11.1 – Permutations (Part 2) Permutations: _________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ For example, “How many ways can you arrange 4 people in a line” is a permutation, whereas “How many ways can you choose a group of 4 from your math class” is not. Example #1

A group of 6 friends need to sit on a bench so that they can have a group picture taken How many ways can these 6 friends be arranged on the bench? Example #2

A librarian would like to arrange 3 books in a display case for her students. She has 7 books to choose from. How many different ways can she arrange the books? Example #3

A basketball coach must have a starting line-up of 5 players. The coach has 12 basketball players on his team. How many ways can the coach select his starting line-up if all players on the team can play any of the 5 starting positions?

P1 P2 P3

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Permutations with Repetitions Example #4

How many different words can be formed using the letters: a) SEA b) SEE

In general, a set of “𝑛” objects with “𝑎” objects that are identical can be arranged in 𝑛!

𝑎!

different ways. And, in general a set of “𝑛” objects with “𝑎” objects that are identical, “𝑏” other identical

objects and “𝑐” other identical objects can be arranged in 𝑛!

𝑎!𝑏!𝑐! different ways.

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Example #5

How many different arrangements can you make suing all the letters in MANITOBA? Example #6

How many different permutations of the word MISSISSIPPI are there? Example #7

How many different arrangements of all the numbers of 81818 can you make? Example #8

How many different paths can you follow from A to B if you can only move up and right?

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Permutations with Constraints Example #9

Five people (A, B, C, D, and E) are seated on a bench. In how many way can they be arranged if a) E must be seated in the middle? b) A and B must be seated together? c) A and B must NOT sit together? d) A and C must sit together and B and D must sit together?

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Example #10

Congratulations, you are now in University, you will now need to arrange the textbooks on your bookshelf. You have to arrange 3 calculus books, 5 biology books and 4 economics books on the shelf. a) How many ways can you arrange the books if there are no restrictions? b) How many ways can you arrange the books if they must be arranged by subject area? c) How many ways can you arrange the books if all the calculus books must go on the right end of the shelf?

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Permutations with Cases Every example we’ve seen so far is a permutation with one case. But, sometimes it’s impossible to summarize all possible permutations into a single equation. If you cannot summarize a situation, you must break that situation into distinct cases and add the resulting permutations for each case to find the total. Example #11

The letters ARTEFQU are to be arranged in a row. But, the arrangement must begin and end with both consonants (RTFQ) or both vowels (AEU). How many ways can this be done? CASE 1: Begin and end with consonants CASE 2: Begin and end with vowels Therefore, there are _____________ +_____________=___________________ possible permutations.

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Example #12

How many different 3-digit even numbers greater than 301 can you make using the numbers 1, 2, 3, 4, 5, and 6? No digits are repeated. CASE 1: ______________________ CASE 2: ______________________ CASE 3: ______________________ Example #13

Using the numbers 1, 2, 3, 4, 5, and 6, how many numbers less than 300 are possible?

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Chapter 11: PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM

11.2 – Combinations Combination: __________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ For example, if you were to form a group of three from students A, B, C, and D. The only combinations are ABC, ABD, ACD and BCD. Note that ABC is the same as ACB or BCA or BAC or CAB or CBA – those permutations all the same combination or group. If you have “𝑛” distinct items and “𝑟” items are taken BUT NOT ARRANGED, then the number of combinations can be found using

Example #1

There are four books on a shelf. In how many ways can you select two books?

P1 P2 P3

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Example #2

In the Lotto 649, the object is to guess the 6 winning numbers out of 49 possible choices. The order of the picks is not important. How many possible winning combinations are there? Example #3

There are 14 females and 8 males in a math class. You want to form a group of 5. a) How many groups can be formed? b) How many groups can be formed with 2 females and 3 males? c) How many groups can be made with Joseph, one other male, and 3 females?

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Example #4

The chess club has 9 grade 10’s, 5 grade 11’s, and 14 grade 12’s. The clubs coach must select a team of 2 students from each grade to compete in a tournament. How many possible teams could he select? Combinations with Case Example #5

a) (Non-Case Example) On a test, you must answer 3 of the 5 short answer questions and 2 of the 6 essay questions. How many combinations of questions are possible? b) (Case Example) On a test, you must answer at least 4 of the 5 short answer questions and 2 of the 6 essay questions. How many combinations of questions are possible?

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Example #6

A bag contains 6 red hockey pucks and 8 black hockey pucks. Assuming the pucks are distinguishable, draw 5 pucks from the bag. How many possible ways are there to have at least 3 red pucks? Example #7

A committee of four students is to be selected from 4 girls and 5 boys. In how many ways can the selection be made if a) There must be more girls than boys on the committee? b) There must be at least one girl on the committee?

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Chapter 11: PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM

Permutations & Combinations (Arithmetic) Recall: A Permutation is an arrangement of objects in which the order is important. It is the number of ways “n” objects can be placed into “r” positions. If you have “𝑛” distinct items and “𝑟” items are taken and arranged, then the number of permutations can be found using: Recall: A Combination is a selection of a group of objects from a larger group, but where the order is not important. It is the number of combinations of “n” items taken “r” at a time. If you have “𝑛” distinct items and “𝑟” items are taken BUT NOT ARRANGED, then the number of combinations can be found using

P1 P2 P3

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Example #1

Solve each of the following.

a) 𝑃2𝑛 = 56 b) 𝑃3 = 120𝑛

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Example #2

Simplify each of the following

a) 𝐶𝑛 5

𝐶𝑛−1 3 b) ( 𝐶𝑛−1 3) (

1

𝐶𝑛−2 3)

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Example #3

Solve each of the following

a) 2( 𝐶𝑛 2) = 𝐶𝑛+1 3 b) 720 ( 𝐶𝑛 5) = 𝑃𝑛+1 5

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Chapter 11: PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM

11.3 – Binomial Theorem Let’s expand some binomials! (𝑎 + 𝑏)1 = 𝑎 + 𝑏 (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏2 (𝑎 + 𝑏)3 = 𝑎3 + 3𝑎2𝑏 + 3𝑎𝑏2 + 𝑏3 (𝑎 + 𝑏)4 = 𝑎4 + 4𝑎3𝑏 + 6𝑎2𝑏2 + 4𝑎𝑏3 + 𝑏4 When we have (𝑎 + 𝑏)𝑛 𝑎 = first term of the binomial 𝑏 = second term of the binomial 𝑛 = exponent of the binomial Checking the patterns above, we find: - The exponent on 𝒂 _______________ by 1 in each term of the expansion and the exponent on 𝒃 _______________ by 1 - There are always _______________ terms of the expansion (1 more than the value of the exponent. - The _______________ of the exponents of each individual term of the expansions is equal to the exponent of the binomial. - The coefficients are symmetric (read the same left to right) and can be found using combinations where 𝒏 is the value of the exponent. For example, the coefficients of (𝑎 + 𝑏)4 are:

𝐶4 0 = 1 𝐶4 1 = 4 𝐶4 2 = 6 𝐶4 3 = 4 𝐶4 4 = 1

P4

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Binomial Theorem: _____________________________________________________ ______________________________________________________________________

d(𝑎 + 𝑏)𝑛 = 𝐶𝑛 0(𝑎)𝑛(𝑏)0 + 𝐶𝑛 1(𝑎)

𝑛−1(𝑏)1 + 𝐶𝑛 2(𝑎)𝑛−2(𝑏)2 + ⋯ + 𝐶𝑛 𝑛−1(𝑎)

1(𝑏)𝑛−1 + 𝐶𝑛 𝑛(𝑎)0(𝑏)0

Example #1

Expand (𝑥 + 𝑦)6 using the binomial theorem.

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Pascal’s Triangle

1 1 1

1 2 1 1 3 3 1

1 4 6 4 1 1 5 10 10 5 1

1 6 15 20 15 6 1 1 7 21 35 35 21 7 1

a) Determine the next line of the triangle b) Find the connection between Pascal’s Triangle and the Binomial Theorem. c) Determine the numeric coefficient of the 6th term of the binomial expansions of (𝑥 + 𝑦)7. d) Explain a situation which Pascal’s Triangle does not give the coefficient of each term of the expansion.

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The coefficients in a binomial expansion can be determined from Pascal’s triangle. In the expansion (𝑥 + 𝑦)𝑛, where 𝑛 ∈ Ν, the coefficients of the terms are identical to the numbers in the (𝑛 + 1)th row of Pascal’s triangle. The coefficients in a binomial expansion can also be determined using combinations (i.e. Binomial Theorem). General Term of a Binomial Expansion To find a specific term:

𝑡𝑘+1 = 𝐶𝑛 𝑘(𝑎)𝑛−𝑘(𝑏)𝑘

Where: 𝑎 = first term of the binomial 𝑏 = second term of the binomial 𝑛 = exponent of the binomial We use 𝒌 to find a specific term. For example, if we are asked to find the 9th term of a binomial expansion, we use… 𝑘 + 1 = 9 → 𝑘 = 8

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Example #2

Determine the 8th term in the binomial expansion of (𝑥 + 𝑦)25 Example #3

Determine the 10th term in the binomial expansion of (𝑥 − 2)12 Example #4

Determine the 4th term in the binomial expansion of (𝑥 +1

2𝑥2)

9

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Finding a Term containing a certain expression Example #5

Find the term that contains 𝑥20 in the binomial expansion of (2𝑥 − 𝑥4)14

• What is 𝑎 =

• What is 𝑏 =

• What is 𝑛 = - We know that the term we are looking for is 𝑡𝑘+1, so 𝑡𝑘+1 = _________ - We know that this term will have 𝑥20 as its veriable - The numeric coefficient does not make a difference.

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Example #6

Find the term that contains 𝑥4, when simplified, of the binomial expansion of (𝑥2 −2

𝑥)11

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Example #7

Determine the constant term in the binomial expansion of (3𝑝 −1

𝑝)

18

.

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Example #8

Determine the value of 𝑛 if the 5th term in the binomial expansion of (2𝑥2 −1

𝑥)

𝑛

is 280𝑥2