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Math 1313 Section 5.4 Permutations and Combinations

Definition: n-Factorial For any natural number n, 123)2)(1(! nnnn . Note: 0! = 1 Example: 21.2!2 ; 1201.2.3.4.5!5 .

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Arrangements: n different objects can be arranged in !n different ways. Example: Arrange the letters: ABC

1) ABC 2) ACB 3) BAC 4) BCA 5) CAB 6) CBA

3 different letters can be arranged in 3!=6 different ways. Example: In how many different ways can you arrange 4 different books on a shelf? Example: In how many ways can 5 people be seated in a row of 5 chairs? Example: How many 4 letter words can you form by arranging the letters of the word MATH?

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Example: Arrange the letters: ABB

1) ABB 2) ABB 3) BBA 4) BBA 5) BAB 6) BAB

We do not get 6 different arrangements! Arrangements of n objects, not all distinct: Given a set of n objects in which 1n objects are alike and of one kind, 2n objects are alike and of another kind,…, and, finally, rn objects are alike and of yet another kind so that nnnn r ...21 , then the these objects can be arranged in:

!!!

!

21 rnnn

n

different ways.

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Example: In how many ways can you arrange 2 red, 4 blue and 5 black pencils? Example: How many 8 letter words can you form by arranging the letters of the word MATHEMATICS? Example: How many 11-letter words can you form by arranging the letters of the word MISSISSIPPI?

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POPPER for Section 5.4 Question# : In how many ways can you create a 4-letter word using all the letters in “MALL”?

A) 24 B) 120 C) 12 D) 6 E) 16 F) None of the above

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Permutations of n elements taken r at a time: ( , )P n r or n rP When finding permutations, ORDER is important. Definition: A permutation is an arrangement of a specific set where the order in which the objects are arranged is important.

Formula: P(n, r) = )!(

!

rn

n

, r < n

where n is the number of distinct objects and r is the number of distinct objects taken r at a time.

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Example: In howmany different ways can you create a two-letter word using the letters “ABC”? Answer: P(3,2)

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Example: A shelf has space for 4 books but you have 6 books. How many different arrangements can you make?

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Combinations of n objects taken r at a time: ( , )C n r or n rC When finding combinations, order does NOT matter. Definition: A combination is an arrangement of a specific set where the order in which the objects are arranged is not important.

Formula: C(n, r))!(!

!

rnr

n

, r < n

where n is the number of distinct objects and r is the number of distinct objects taken r at a time.

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Example: In how many ways can you choose 2 letters from the word “ABC”? Answer: C(3,2)

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Example: A group of 4 students will be chosen from a class of 6 students to represent the group in a meeting. How many different groups can be chosen?

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MISCELLANEOUS EXAMPLES about arrangements, permutations and combinations: Example 1: In how many ways can 7 cards be drawn from a well-shuffled deck of 52 playing cards? Example 2: An organization has 20 members. In how many ways can the positions of president, vice-president, secretary, treasurer, and historian be filled if not one person can fill more than one position?

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Example 3: In how many ways can 10 people be assigned to 5 seats? Example 4: An organization needs to make up a social committee. If the organization has 25 members, in how many ways can a 10 person committee be made?

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Example 5: Seven people arrive at a ticket counter at the same time to buy concert tickets. In how many ways can they line up to purchase their tickets?

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Example 6: A coin is tossed 3 times.

In how many outcomes do exactly 2 heads occur?

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Example 7: A coin is tossed 5 times.

a. In how many outcomes do exactly 3 heads occur? { (H1 H 2 H 3 TT), ( H1 H 2 T H 4 T), ( H1 H 2 TT H 5 ), (H1 TH 3 T H 5 ), (H1 T TH 4 H 5 ), (H1 T H 3 H 4 T), ( TH 2 H 3 H 4 T), (T H 2 H 3 T H 4 ), (TH 2 TH 4 H 5 ), (T TH 3 H 4 H 5 ) } b. In how many outcomes do at least 4 heads occur?

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Example 8: A coin is tossed 18 times. a. In how many outcomes do exactly 7 tails occur? b. In how many outcomes do at most 2 tails occur? c. In how many outcomes do at most 16 heads occur? d. In how many outcomes do at least 3 tails occur?

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Example 9: A student belongs to a reading club. This month he must purchase 2 novels and 4 history books. If there are 10 novels and 12 history books to choose from, in how many ways can he choose his 6 purchases?

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Example 10: A committee of 16 people, 7 women and 9 men, is forming a subcommittee that is to be made up of 6 women and 4 men. In how many ways can the subcommittee be formed?

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Example 11: In how many ways can 5 spades be chosen if 8 cards are chosen from a well-shuffled deck of 52 playing cards?

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Example 12: A customer at a fruit stand picks a sample of 6 avocados at random from a crate containing 24 avocados of which 8 are rotten. a. In how many ways can the batch contain 5 rotten avocados? b. In how many ways can the batch contain at most 2 rotten avocados?

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Example 13: A computer store receives a shipment of 25 printers; including 5 that are defective. 6 printers are chosen to be displayed in the store. How many selections can be made? How many selections contain 2 defective printers? How many selections contain at least 4 defective printers? How many selections contain at least 1 defective printer?

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Example 14: A committee of 6 teachers will be chosen from 10 math and 12 science teachers. In how many ways can this group be chosen if: a. there should be 4 math teachers in this group? b. there should be at least 4 math teachers in this group? c. there should be at most 2 math teachers in this group?

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POPPER for Section 5.4 Question# : There is a group of 6 math and 2 science teachers. In how many ways can you form a committee of 4 math and 1 science teachers from this group?

a) C(8,4) b) C(8,5) c) C(6,4) d) C(6,4)*C(2,1) e) C(8,4)*C(2,1) f) None of these

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Exercises: 1. A crate of 17 apples contains 4 rotten. Seven of these apples are chosen. a. How many selections contain at least 3 rotten apples? 3,146 b. How many selections contain at least 3 good apples? 286 2. A 9-person committee is to be selected out of 2 departments, A and B, with 14 and 17 people, respectively. a. In how many ways can fewer than 3 people from department A be selected? 2,134,418 b. In how many ways can no more than 4 people from department B be selected? 7,326,605 c. In how many ways can at least 7 people from department A be selected? 519,805 3. A bag contains 15 marbles, 3 yellow, 4 white and 8 blue. Five are chosen at random. a. In how many ways can more than 3 blue marbles be chosen? 546 b. In how many ways can less than 4 white be chosen? 2,992 4. Eight cards are chosen from a well-shuffled deck of 52 playing cards. a. In how many ways can at least 2 face cards be chosen? b. In how many ways can at most 5 red cards be chosen? c. In how many ways can at least 3 diamonds be chosen? d. In how many ways can less than 7 spades be chosen? e. In how many ways can the eight cards have the same suit? f. In how many ways can the eight cards have the same color?