Permutations & Combinations

1. Chapter 13 sec. 3 Permutations & Combinations

2. Permutations • Def. • Is an ordering of distinct objects in a straight line. If we select r different objects from a set of n objects and arrange them in a straight line, this is called permutation of n objects taken r at a time. • Order matters!!!!! • Denoted by P(n,r)

3. Meaning what? • What does P(5,3) mean? • n is the number of objects from which you may select. • r is the number of objects that you are selecting. • That you are counting permutations formed by 3 different objects from a set of five available objects.

4. Example • How many permutations are there of the letters z, r, t, and w. Write the answer in P(n,r) notation. • Solution: • One way is to make a list. (too long.) • Using the slot diagram.

5. Slot diagram method • Without repetition, there are 4 letters which can be for the first position, 3 for the second, and so on. 1st letter 2nd 3rd 4th x xx Therefore P(4,4) = 24 permutations. 4 3 2 1

6. Try this! • Find the number of permutations. Write it as P(n,r) notation. • Eight objects taken three at a time. Questions to think about. 1. How many objects (n)? 2. The number of objects being selected (r)?

7. Solution • There are 8 objects which is n. • 3 objects are being selected. (r) • P(8,3) = 8 X 7 X 6 = 336

8. Factorial Notation • n!, called n factorial • n•(n-1)•(n-2)•∙∙∙•2•1 • 0!=1

9. Example • 6! = 6x5x4x3x2x1 = 720 • (6-3)! = 3! = 3x2x1 = 6 • 3!/4! =(3x2x1)/(4x3x2x1) = 1/4

10. Why do you need this? • To help you compute P(n,r)! • P(n,r) = n! /(n-r)!

11. Examples • Find the Permutation • A) 9 objects taken 4 at a time. • B) 20 objects taken 7 at a time. • C) 5 objects taken 2 at a time.

12. Solution • A) P(9,4) = 9!/5! = 9x8x7x6=3024 • B) P(20,7) = 20!/13!= 20x19x18…x14 = 390,700,800 • C) P(5,2) = 5!/3! = 20

13. Combination • Def. • If we choose r objects from a set of n objects, we say that we are forming a combination of n objects taken r at a time. • Notation C(n,r) = P(n,r) / r! = n! / [r!(n-r)!]

14. Meaning What?!!! • We are only concerned only with choosing a set of elements, but the order of the elements is not important.

15. Meaning What!@##?! 2 • This means that if the permutations number is big, the combination number will be smaller.

16. Examples • Find the Combinations • A) Eight objects taken three at a time. • B) Nine objects taken six at a time. • C) How many 3 elements sets can be chosen from a set of 5 objects.

17. Solutions • A) C(8,3) = 8!/(3!5!) = 8x7x6/6= 56 • B) C(9,6) = 9!/(6!3!) =9x8x7/3x2=84 • C) C(5,3) = 10

18. Problems • In the game of poker, five cards are drawn from a standard 52-card deck. How many different poker hands are possible? • Solution: • C(52,5) = 2, 598, 960

19. Give your answers using P(n,r) or C(n,r) notation. The key is if order matters or not. • 1. Annette has rented a summer house for next semester. She wants to select four roommates from a group of six friends.

20. 2. There are 7 boats that will finish the America’s Cup yacht race. • 3. A bicycle lock has three rings with the letters A through K on each ring. To unlock the lock, a letter must be selected on each ring. Duplicate letters are not allowed, and the order in which the letters are selected on the rings does not matter.

21. Solution • C(6,4) • P(7,7) • C(11,3)