1. Chapter 12 – Probability and Statistics 12.2 – Permutations and Combinations

2. Today we will learn how to: Solve problems involving permutations Solve problems involving combinations

3. 12.2 – Permutations and Combinations Permutation – when a group of objects or people are arranged in a certain (particular) order The order of the objects is very important Linear permutation – the arrangement of objects or people in a line The number of permutations of n distinct objects taken r at a time is given by P (n, r) = n! (n – r)! n P r

4. 12.2 – Permutations and Combinations Example 1 Eight people enter the Best Pie contest. How many ways can blue, red, and yellow ribbons be awarded?

5. 12.2 – Permutations and Combinations Suppose you want to rearrange the letters of the word geometry to see if you can make a different word. If the two es were not identical, the eight letters in the word could be arranged in P(8, 8) ways. To account for the identical es, divide P(8, 8) by the number of arrangements of e. The two es can be arranged in P(2, 2) ways. P(8, 8) = 8! P(2, 2) 2! = 8 · 7 · 6 · 5 · 4 · 3 · 2! or 20,160 2!

6. 12.2 – Permutations and Combinations Permutations with Repetitions The number of permutations of n objects of which p are alike and q are alike is n! p! q!

7. 12.2 – Permutations and Combinations Example 2 How many different ways can the letters of the word BANANA be arranged?

8. 12.2 – Permutations and Combinations Combinations – an arrangement or selection of objects in which the order is not important The number of combinations of n objects taken r at a time is written C(n, r) or n C r You know that there are P(n, r) ways to select r objects from a group of n if the order is important. There are r! ways to order the r objects that are selected, so there are r! permutations that are all the same combination. Therefore: C(n, r) = P(n, r) or n!. r! (n – r)! r!

9. 12.2 – Permutations and Combinations Combinations The number of combinations of n distinct objects taken r at a time is given by C (n, r) = n!. (n – r)! r!

10. 12.2 – Permutations and Combinations Example 3 Five cousins at a family reunion decide that three of them will go to pick up a pizza. How many ways can they choose the three who will go?

11. 12.2 – Permutations and Combinations In more complicated situations, you may need to multiply combinations and/or permutations

12. 12.2 – Permutations and Combinations Example 4 Six cards are drawn from a standard deck of cards. How many hands consist of two hearts and four spades?

13. 12.2 – Permutations and Combinations HOMEWORK Page 693 #11 – 37 odd