COUNTING & PROBABILITY 1

Transcription

1 COUNTING & PROBABILITY 1 1) A restaurant offers 7 entrees and 6 desserts. In how many ways can a person order a two-course meal? 2) In how many ways can a girl choose a two-piece outfit from 5 blouses and 6 skirts? 3) A restaurant offers a choice of 4 salads, 5 main courses, and 4 desserts. How many possible meals are there? 4) An apartment complex offers apartments with four different options, designated by A through D. 1) 2) 3) 4) A = number of bedrooms (one through four) B = number of bathrooms (one through three) C = floor (first through fifth) D = outdoor additions (balcony or no balcony) How many apartment options are available? 5) A person can order a new car with a choice of 6 possible colors, with or without air conditioning, with or without heated seats, with or without anti-lock brakes, with or without power windows, and with or without a CD player. In how many different ways can a new car be ordered in terms of these options? 6) You are taking a multiple-choice test that has 7 questions. Each of the questions has 5 choices, with one correct choice per question. If you select one of these options per question and leave nothing blank, in how many ways can you answer the questions? 7) License plates in a particular state display 2 letters followed by 2 numbers. How many different license plates can be manufactured? (Repetitions are allowed.) 8) How many different four-letter secret codes can be formed if the first letter must be an S or a T? 5) 6) 7) 8) 9) Jamie is joining a music club. As part of her introductory package, she can choose from 12 rock selections, 10 alternative selections, 7 country selections and 5 classical selections. If Jamie chooses one selection from each category, how many ways can she choose her introductory package? 9) 1

2 10) There are 8 performers who are to present their acts at a variety show. How many different ways are there to schedule their appearances? 11) There are 7 performers who are to present their acts at a variety show. One of them insists on being the first act of the evening. If this request is granted, how many different ways are there to schedule the appearances? 12) You want to arrange 8 of your favorite CD's along a shelf. How many different ways can you arrange the CD's assuming that the order of the CD's makes a difference to you? 13) A teacher and 10 students are to be seated along a bench in the bleachers at a basketball game. In how many ways can this be done if the teacher must be seated in the middle and a difficult student must sit to the teacher's immediate left? 10) 11) 12) 13) Evaluate the factorial expression. 14) 14) 15) 7! - 4! 15) 16) 12! 3 16) 17) ( 9-5)! 17) Use the formula for to solve. 18) A church has 10 bells in its bell tower. Before each church service 3 bells are rung in sequence. No bell is rung more than once. How many sequences are there? 19) A club elects a president, vice-president, and secretary-treasurer. How many sets of officers are possible if there are 13 members and any member can be elected to each position? No person can hold more than one office. 20) In a contest in which 8 contestants are entered, in how many ways can the 5 distinct prizes be awarded? 18) 19) 20) 2

3 21) How many arrangements can be made using 2 letters of the word HYPERBOLAS if no letter is to be used more than once? Solve the problem. 22) In how many distinct ways can the letters in IMMUNOLOGY be arranged? 23) A signal can be formed by running different colored flags up a pole, one above the other. Find the number of different signals consisting of 7 flags that can be made if 4 of the flags are white, 2 are red, and 1 is blue. 21) 22) 23) 24) In how many distinct ways can a 9-digit number be made using four 5 s and five 4 s? 24) In the following exercise, does the problem involve permutation or combination? Explain your answer. It is not necessary to solve the problem. 25) A record club offers a choice of 7 records from a list of 45. In how many ways can a member make a selection? 25) A) Permutations, because the order of the records selected does matter. B) Combinations, because the order of the records selected does not matter. Solve the problem. 26) From 10 names on a ballot, a committee of 4 will be elected to attend a political national convention. How many different committees are possible? 26) A) 151,200 B) 5040 C) 2520 D) ) To win at LOTTO in a certain state, one must correctly select 6 numbers from a collection of 55 numbers (one through 55). The order in which the selections are made does not matter. How many different selections are possible? 27) A) 720 B) 28,989,675 C) 25,827,165 D) ) In how many ways can a committee of three men and four women be formed from a group of 11 men and 11 women? A) 554,400 B) 7,840,800 C) 54,450 D) ) 3

4 29) A physics exam consists of 9 multiple-choice questions and 6 open-ended problems in which all work must be shown. If an examinee must answer 5 of the multiple-choice questions and 2 of the open-ended problems, in how many ways can the questions and problems be chosen? 29) A) 1890 B) 453,600 C) 540 D) 261,273,600 In the following exercises, does the problem involve permutations or combinations? Explain your answer. It is not necessary to solve the problem. 30) One hundred people purchase lottery tickets. Three winning tickets will be selected at random. If first prize is $100, second prize is $50, and third prize is $25, in how many different ways can the prizes be awarded? 30) A) Combinations, because the order of the prizes awarded does not matter. B) Permutations, because the order of the prizes awarded matters. 31) How many different user ID's can be formed from the letters W, X, Y, Z if no repetition of letters is allowed? 31) A) Permutations, because the order of the letters matters. B) Combinations, because the order of the letters does not matter. 32) Five of a sample of 100 computers will be selected and tested. How many ways are there to make this selection? 32) A) Combinations, because the order of the computers selected does not matter. B) Permutations, because the order of the computers selected does matter. 4

5 ANSWERS 1) 42 2) 30 3) 80 4) 120 5) 192 6) 78,125 7) 67,600 8) 35,152 9) ) 40,320 11) ) 40,320 13) 362,880 14) ) 5,016 16) 24 17) 24 18) ) 1,716 20) 6,720 21) 90 22) 907,200 23) ) ) B 26) D 27) B 28) C 29) A 30) B 31) A 32) A 5