Math 1313 Section 5.4 Permutations and Combinations

Transcription

1 Math 1313 Section 5.4 Permutations and Combinations Definition: n-factorial For any natural number n, n! n( n 1)( n 2) Note: 0! = 1 Example: 2! ; 5!

2 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? 2

3 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 n 1 objects are alike and of one kind, n 2 objects are alike and of another kind,, and, finally, n r objects are alike and of yet another kind so that n n... n 1 2 r n, then the these objects can be arranged in: different ways. n! n! n! n 1 2 r! 3

4 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? 4

5 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 5

6 Permutations of n elements taken r at a time: Pnr (, ) or n P r 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) = n!, r < n ( n r)! where n is the number of distinct objects and r is the number of distinct objects taken r at a time. 6

7 Example: In howmany different ways can you create a two-letter word using the letters ABC? Answer: P(3,2) 7

8 Example: A shelf has space for 4 books but you have 6 books. How many different arrangements can you make? 8

9 Combinations of n objects taken r at a time: Cnr (, ) or n C 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. n! Formula: C(n, r), r < n r!( n r)! where n is the number of distinct objects and r is the number of distinct objects taken r at a time. r 9

10 Example: In how many ways can you choose 2 letters from the word ABC? Answer: C(3,2) 10

11 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? 11

12 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? 12

13 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? 13

14 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? 14

15 Example 6: A coin is tossed 3 times. In how many outcomes do exactly 2 heads occur? 15

16 Example 7: A coin is tossed 5 times. a. In how many outcomes do exactly 3 heads occur? { (H 1 H 2 H 3 TT), ( H 1 H 2 T H 4 T), ( H 1 H 2 TT H 5 ), (H 1 TH 3 T H 5 ), (H 1 T TH 4 H 5 ), (H 1 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? 16

17 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? 17

18 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? 18

19 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? 19

20 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? 20

21 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? 21

22 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? 22

23 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? 23

24 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 24

25 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? 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, 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, 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? 25