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Permutations and Combinations

Permutations and Combinations. Alg 1 B. Factorials. What is a factorial 5! For example Try 9! 18!. Permutations vs. Combinations. Both are ways to count the possibilities The difference between them is whether order matters or not Consider a poker hand: A ♦, 5♥, 7♣, 10♠, K♠

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Permutations and Combinations

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  1. Permutations and Combinations Alg 1 B

  2. Factorials What is a factorial 5! For example Try 9! 18!

  3. Permutations vs. Combinations • Both are ways to count the possibilities • The difference between them is whether order matters or not • Consider a poker hand: • A♦, 5♥, 7♣, 10♠, K♠ • Is that the same hand as: • K♠, 10♠, 7♣, 5♥, A♦ • Does the order the cards are handed out matter? • If yes, then we are dealing with permutations • If no, then we are dealing with combinations

  4. Permutations • A permutation is an ordered arrangement of the elements of some set S • Let S = {a, b, c} • c, b, a is a permutation of S • b, c, a is a different permutation of S • An r-permutation is an ordered arrangement of r elements of the set • A♦, 5♥, 7♣, 10♠, K♠ is a 5-permutation of the set of cards • The notation for the number of r-permutations: P(n,r) • The poker hand is one of P(52,5) permutations

  5. Permutations • Number of poker hands (5 cards): • P(52,5) = 52*51*50*49*48 = 311,875,200 • Number of (initial) blackjack hands (2 cards): • P(52,2) = 52*51 = 2,652 • r-permutation notation: P(n,r) • The poker hand is one of P(52,5) permutations

  6. r-permutations example • How many ways are there for 5 people in this class to give presentations? • There are 27 students in the class • P(27,5) = 27*26*25*24*23 = 9,687,600 • Note that the order they go in does matter in this example!

  7. 2nd permutation example • How many permutations of {a, b, c, d, e, f, g} end with a? • Note that the set has 7 elements • The last character must be a • The rest can be in any order • Thus, we want a 6-permutation on the set {b, c, d, e, f, g} • P(6,6) = 6! = 720 • Why is it not P(7,6)?

  8. Combinations • What if order doesn’t matter? • In poker, the following two hands are equivalent: • A♦, 5♥, 7♣, 10♠, K♠ • K♠, 10♠, 7♣, 5♥, A♦ • The number of r-combinations of a set with n elements, where n is non-negative and 0≤r≤n is:

  9. Combinations example • How many different poker hands are there (5 cards)? • How many different (initial) blackjack hands are there?

  10. Decide which and then Solve • You have 14 CD’s . You can arrange 12 of the 14 in a CD wallet. In how many ways can you arrange the CD’s in the wallet?

  11. Decide which and then Solve You go to Applebee’s and order the choose 2 deal. With this deal you can choose 2 entrees from a list of 8. How many ways can you do this?

  12. Decide which and then Solve • In English class for your final exam you must choose to respond to 5 essay questions out of a list of 8. How many answer keys must the teacher make?

  13. Classwork(please work to the bell) • Pg. 854 12-26 even, 30, 32, 33 • Pg. 858 6-14 even, 15, 16-19, Bonus 23

  14. Quick survey • I felt I understood the material in this slide set… • Very well • With some review, I’ll be good • Not really • Not at all

  15. Quick survey • The pace of the lecture for this slide set was… • Fast • About right • A little slow • Too slow

  16. Quick survey • How interesting was the material in this slide set? Be honest! • Wow! That was SOOOOOO cool! • Somewhat interesting • Rather boring • Zzzzzzzzzzz

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