2. Honors PreCalculus: Section 9.1 Basic Combinatorics

3. Fundamental Counting Principal The total number of ways in which a series of successive “events” can occur is found by multiplying the number of ways in which each individual “event” can occur.

4. Sally is packing for vacation. She wants to maximize her wardrobe options. She plans on taking the following clothing items. How many possible outfits will she have on her trip? jeans – 2 pairs, shorts – 3 pairs, skirts – 2; shirts – 9; shoes – 4 pairs 7 bottoms Number of Outfits = Example 1 252 7 9 4 =

5. Example 2: You are taking a multiple choice test that has five questions. Each of the questions has five possible answers with one correct answer per question. If you select one answer per question how many ways can you answer the questions? Q1 Q2 Q3 Q4 Q5 5 5 5 5 5 = 5 5 Example 3 Telephone numbers in the U.S. begin with a 3 digit area code followed by a seven digit local number. Area codes and local numbers cannot begin with a 0 or 1. How many different telephone numbers are possible? Possible numbers per digit 8 10 10 8 10 10 10 10 10 10 = 6,400,000,000 area code local number 88 10 10 10 10

6. Permutations An ordered arrangement that occurs when…. No item is used more than once. The order of the arrangement makes a difference.

7. Example 4 You are the coach of a little league baseball team with 13 players. You need to put together the batting order with nine players. How many batting orders are possible? Bat1Bat 2Bat 3Bat 4Bat 5Bat 6Bat 7Bat 8Bat 9 13 choices 11 choices 10 choices 9 choices 8 choices 7 choices 6 choices 5 choices 13 12 1110 9 8 7 6 5 =259,459,200 12 choices

8. Permutations of n Things Taken r at a Time The number of possible permutations if r items are taken from n items is: n P r = ______ n! (n-r)! Lets go back to our baseball batting order n = 13 r = 3 13 P 9 = _____ = __________________________ = 259,459,200 13! 13  12  11  10  9  8  7  6  5  4! 4!

9. Example 5 A corporation has seven members on its board of directors. In how many different ways can it select a president, vice-president, secretary and treasurer? 840

10. Combinations An arrangement that occurs when…. The items are selected from the same group. No item is used more than once. The order of the items makes no difference.

11. Distinguishing between Permutations & Combinations Do you see the difference between a permutation and a combination? A permutation involves situations in which order matters. A combination involves situations in which the order of the items makes no difference.

12. Example 6: Distinguishing between Permutations and Combinations 1. Six Students are running for student council president, vice-president and treasurer. The student with the greatest number of votes becomes the president, the student with the second greatest number of votes becomes the vice-president and the student with the third greatest number becomes the treasurer. How many different outcomes are possible for these three positions? Permutation or Combination 2. Six people are on the board of supervisors for the park district. A three person committee is needed to study the possibility of expanding the park. How many different committees could be formed from the 6 people? Permutation or Combination

13. Example 6: Distinguishing between Permutations and Combinations 3. Baskin Robbins offers 31 different flavors of ice cream. One of their items is a bowl consisting of three scoops of ice cream, each a different flavor. How many such bowls are possible? Permutation or Combination 4. In a race in which there are 50 runners and no ties, in how many different ways can the first three finishers come in? Permutation or Combination

14. Combinations of n Things Taken r at a Time The number of possible combinations if r items are taken from n items is: n c r = ______ n! (n-r)!r!

15. Example 7 From a group of 10 physicians, how many ways can 4 doctors be selected to attend a conference on accupuncture? 210

16. Example 8: A person playing poker is dealt a 5 card hand from a standard deck of 52 cards. The order in which the 5 cards are dealt does not matter. How many different 5 card poker hands are possible? = 2,598,960 Homework pages 708 -709 problems 1 – 33 odds