Section : Combinations and Permutations


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1 Section : Combinations and Permutations Diana Pell A construction crew has three members. A team of two must be chosen for a particular job. In how many ways can the team be chosen? How many words (strings of letters) of two distinct letters can be formed from the letters {a, b, c}. Factorial 1
2 Combinations (Order does not matter!) Combination Formula: The number of combinations of n objects taken r at a time is Exercise 1. a) C(100, 2) C(n, r) = n! (n r)!r! b) C(5, 5) Exercise 2. The board of directors of a corporation has 10 members. In how many ways can they choose a committee of 3 board members to negotiate a merger? Exercise 3. How many different selections of two books can be made from a set of nine books? 2
3 Permutations (Order matters!) Permutation Formula: The number of permutations of n objects taken r at a time is P (n, r) = n(n 1)(n 2)(n 3) (n r + 1) OR P (n, r) = n! (n r)! Exercise 4. Eight horses are entered in a race in which a first, second, and third prize will be awarded. Assuming no ties, how many different outcomes are possible? Exercise 5. A club has 10 members. In how many ways can they choose a slate of four officers, consisting of a president, vice president, secretary, and treasurer? Exercise 6. How many ways can you arrange 5 of 10 books on a shelf? Exercise 7. Suppose that you own 10 sweaters and are going on a trip. How many ways can you select six of them to leave at home? 3
4 Exercise 8. Of the 20 applicants for a job, 4 will be selected for intensive interviews. In how many ways can the selection be made? Exercise 9. A poker hand consists of 5 cards selected from a deck of 52 cards. a) How many different poker hands are there? b) How many different poker hands consist entirely of aces and kings? c) How many different poker hands consist entirely of clubs? d) How many different poker hands consist entirely of red cards? 4
5 Exercise 10. In how many ways can five mathematics books and four novels be placed on a bookshelf if the mathematics books must stay together? Solving a Permutation Problem with Like Objects Exercise 11. How many different passwords can be made using all the letters in the word Mississippi? Exercise 12. (You Try!) How many different passwords can be made using all the letters in the word Massachusetts? 5
6 Exercise 13. A committee has four male and five female members. In how many ways can a subcommittee consisting of two males and two females be selected? Exercise 14. An urn contains 25 numbered balls, of which 15 are red and 10 are white. A sample of 3 balls is to be selected. a) How many different samples are possible? b) How many samples contain all red balls? c) How many samples contain 1 red balls and 2 white balls? d) How many samples contain at least 2 red balls? 6
7 Exercise 15. (You Try!) A fourperson crew for the international space station is to be chosen from a candidate pool of 10 Americans and 12 Russians. How many different crews are possible if there must be at least two Russians? Section 11.3: Basic Concepts of Probability Definition 1. A sample space is a set of all possible outcomes of an experiment. Exercise 16. An experiment consists of flipping a coin once. Find the sample space. Exercise 17. An experiment consist of flipping a coin twice. Find the sample space. Exercise 18. An experiment consists of flipping a coin three times. Find the sample space. 7
8 Exercise 19. An experiment consists of rolling a single die. Find the sample space. Exercise 20. An experiment consists of rolling a green die and a red die. Find the sample space. Definition 2. An event is a subset of a sample space of an experiment. Exercise 21. Suppose and experiment consists of tossing a coin three times and observing the sequence of heads and tails. Determine the event E = exactly two heads. Exercise 22. Suppose that we have two urns  call them urn I and urn II  each containing red balls and white balls. An experiment consists of selecting an urn and then selecting a ball from that urn and noting its color. a) What is a suitable sample space for this experiment? 8
9 b) Describe the event urn I is selected as a subset of the sample space. Definition 3. Experiments in which each outcome has the same probability are said to be experiments with equally likely outcomes. Exercise 23. Experiment consists of flipping a coin two times. Find probability of every outcome in the sample space. Definition 4. If an experiment with sample space S has equally likely outcomes, then for any event E the probability of E is given by P (E) = n(e) n(s) where n(e) and n(s) denote the number of elements in E and S, respectively. Note: Probability is always a number from 0 to 1. Impossible events always have probability 0 and certain events have probability 1. Exercise 24. A single die is rolled. Find the probability of getting a) A 2. 9
10 b) A number less than 5. Exercise 25. Roll a single die. What is the probability that it lands on an odd number? Complement Rule: P (E) = 1 P (E ) Exercise 26. Of the next 32 trials on the docket in a county court, 5 are homicides, 12 are drug offenses, 6 are assaults, and 9 are property crimes. If jurors are assigned to trials randomly, a) what s the probability that a given juror won t get a homicide case? 10
11 b) what s the probability that a juror gets assigned to a case that isn t a drug offense? Exercise 27. (Birthday Problem) A group of five people is to be selected at random. What is the probability that two or more of them have the same birthday? (Assume that each of the 365 days in a year is an equally likely birthday) Exercise 28. A basketball team has four players. What is the probability that at least two of them were born on the same day of the week? Exercise 29. In a random sample of 500 people, 210 had type O blood, 223 had type A, 51 had type B, and 16 had type AB. Set up a frequency distribution and find the probability that a randomly selected person from the general population has 11
12 a) Type O blood. b) Type A or B blood. c) Neither type A nor type O blood. d) A blood type other than AB. 12
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