Slide 1 Math 1520, Lecture 15

Transcription

1 Slide 1 Math 1520, Lecture 15 Formulas and applications for the number of permutations and the number of combinations of sets of elements are considered today. These are two very powerful techniques for counting the number of ways something can happen.

2 Slide 2 Factorial For any natural number n the symbols n! is read n factorial and is computed as n! = n (n 1) (n 2) ! = 1 For example 0! = 1 1! = 1 2! = 2 1 = 2 3! = = 6 4! = = 24 5! = = 120 6! = = 720 7! = = ! = = 40320

3 Slide 3 iclicker Question Find the value of 10! 8! A without a calculator B C. 90 D. No way

4 Answer to Question Find the value of 10! 8! A B C. 90 is the correct answer. D. No way without a calculator

5 Slide 4 Permutations (Part I) A permutation of a set of objects is an arrangement of these objects in a definite order. For example, the permutations of {a, b, c} are a, b, c a, c, b b, a, c b, c, a c, a, b c, b, a The number of permutations of a set of n objects is n! Explain why the answer is n! using the multiplication principle.

6 Slide 5 iclicker Question How many ways can a group of 8 baseball players be arranged in a batting order? A. 8 B. 8! C. 64 D

7 Answer to Question How many ways can a group of 8 baseball players be arranged in a batting order? A. 8 B. 8! is the correct answer. C. 64 D

8 Slide 6 Permutations (Part II) We frequently want to only arrange subsets of the objects in a larger set. For example, how many ways can we arrange 8 baseball players in a line up if they are to be chosen from a group of 13 possible players? For this problem There are 13 possible choices for the leadoff hitter. There are 12 possible choices for the second place hitter.... There are 6 possible choice for the 8th place hitter. So the answer is We can write this answer as 13! 5! = 13! (13 8)! which leads to the general formula on the next slide.

9 Slide 7 Permutations (Part III) The first important formula to remember for this is The number of permutations of n distinct objects taken r at a time is P (n, r) = n! (n r)! 1. For example, how many three letter strings from the word T IGER can be formed? The second important formula to remember for this is a generalization of this formula Given a set of n objects in which n 1 are alike and of one kind, n 2 objects are alike and of another kind,..., and n m object are alike and of yet another kind, so that n 1 + n 2 + n n m = n then the number of permutation of these n objects taken n at a time is n! n 1! n 2! n m! 1. For example, how many distinct ways can the word ANNAGRAM be written?

10 Slide 8 iclicker Question How many ways can 5 people boarding a bus be seated if the bus has eight vacant seats? A. 8! 5 B. 8! 5! C. 8! 3! 8! D. 5! 3!

11 Answer to Question How many ways can 5 people boarding a bus be seated if the bus has eight vacant seats? A. 8! 5 B. 8! 5! C. 8! is the correct answer. 3! 8! D. 5! 3!

12 Slide 9 iclicker Question A firm has 12 inquiries regarding new accounts. How many ways can these accounts be assigned to 4 salespeople, if each salesperson is to handle 3 accounts? 12! A. 4! 3! 12! B. 4! 4! 4! 12! C. 3! 3! 3! 3! 12! D. 4! 4! 4! 3! 3! 3! 3!

13 Answer to Question A firm has 12 inquiries regarding new accounts. How many ways can these accounts be assigned to 4 salespeople, if each salesperson is to handle 3 accounts? 12! A. 4! 3! 12! B. 4! 4! 4! 12! C. is the correct answer. 3! 3! 3! 3! 12! D. 4! 4! 4! 3! 3! 3! 3!

14 Slide 10 Combinations When making permutations, the order is important. If we don t care about the order, we call it a combination and need to use a different formula. The important formula for this situation is as follows. The number of combinations of n distinct objects taken r at a time is given by ( ) n n! C(n, r) = = where r n r r!(n r)! 1. How many ways can a panel of 12 jurors be formed from a pool of 30 prospective jurors? 2. How many five card poker hands are there? 3. Why does this formula work?

15 Slide 11 iclicker A flush in poker is 5 cards of the same suit, such that the cards are not in sequence. For example, A, 2, 3, 4, 5 of hearts and 10, J, Q, K, A of hearts are not flushes because they are straight flushes (or a royal flush). Question How many 5 card heart flushes are there? A. 13! 8! 13! B. 5!8! 13! C. 5!8! 13 13! D. 5!8! 10

16 Answer to Question How many 5 card heart flushes are there? A. 13! 8! 13! B. 5!8! 13! C. 5!8! 13 D. 13! 10 is the correct answer. 5!8!

17 Slide 12 Combinations or Permutations? When looking at a problem, to decide whether to use a combination or a permutation, the key question to ask yourself is: Is the order important? 1. If the order is important then use a permutation. 2. If the order is not important then use a combination.

18 Slide 13 iclicker Question How many different ways can 3 books be chosen from 10 books? A. 10! 3!7! B. 10! 3! C. 10! 7! D. 10! 21

19 Answer to Question How many different ways can 3 books be chosen from 10 books? A. 10! 3!7! B. 10! 3! C. 10! 7! D. 10! 21 is the correct answer.

20 Slide 14 iclicker Question How many different ways can 3 books be chosen from 10 books and lined up on a shelf from left to right? A. 10! 3!7! B. 10! 3! C. 10! 7! D. 10! 21

21 Answer to Question How many different ways can 3 books be chosen from 10 books and lined up on a shelf from left to right? 10! A. 3!7! B. 10! 3! C. 10! 7! D. 10! 21 is the correct answer.