3.4 Additional Topics in Probability and Counting

1. 3.4 Additional Topics in Probability and Counting Arranging objects in order Choosing several objects from a group without regard to order Counting Principles

2. Permutation A permutation is an ordered arrangement of objects. The of different permutations of n distinct objects is n!

3. Try it yourself 1 • Finding the Number of Permutations of n Objects The women’s hockey teams for the 2010 Olympics are Canada, Sweden, Switzerland, Slovakia, United States, Finland, Russia, and China. How many different final standings are possible? 40,320

4. Permutations of n objects taken r at a time The number of permutations of n distinct objects taken r at a time is

5. Try it yourself 2 • Finding Permutations A psychologist shows a list of eight activities to her subject. How many ways can the subject pick a first, second, and third activity? 336

6. Try it yourself 3 • Finding Permutations The board of directors of a company has 12 members. One member is the president, another is the vice president, another is the secretary, and another is the treasurer. How many ways can these positions be assigned? 11,880

7. Distinguishable Permutations The number of distinguishable permutations of n objects

8. Try it yourself 4 • Finding the Number of Distinguishable Permutations A building contractor is planning to develop a subdivision. The contractor wants to plant six oak trees, nine maple trees, and five poplar trees along the subdivision street. The trees are to be spaced evenly. In how many distinguishable ways can they be planted? 77,597,520

9. Combinations of n objects taken r at a time A combination is a selection of r objects from a group of n objects without regard to order and is denoted by The number of combinations of r objects selected from a group of n objects is

10. Try it yourself 5 • Finding the Number of Combinations The manager of an accounting department want to form a three-person advisory committee from the 20 employees in the department. In how many ways can the manager form this committee? 1140

12. Try it yourself 6 • Finding Probabilities A student advisory board consists of 20 members. Two members serves as the board’s chair and secretary. Each member is equally likely to serve in either of the positions. What is the probability of selecting at random the two members who currently hold the two positions? 0.003

13. Try it yourself 7 • Finding Probabilities You have 6 letters consisting of one L, two E’s, two T’s, and one R. If the letters are randomly arranged in order, what is the probability that the arrangement spells the word letter? 0.006

14. Try it yourself 8 • Finding Probabilities Find the probability of being dealt five diamonds from a standard deck of playing cards that also includes two jokers. In this case, the joker is considered to be a wild card that can be used to represent any card in the deck. 0.0009

15. Try it yourself 9 • Finding Probabilities A jury consists of five men and seven women. Three jury members are selected at random for an interview. Find the probability that all three are men. 0.045