The counting principle: permuations and combinations

Permutation and Combinations Handout

Section 1: the counting principle

7. How many sequences of two letters each can be formed?
8. How many three-digit numerals can be written using the symbols 6, 7, and 8?
9. How many different ways can ten questions on a true-false test be answered?
10. An automobile manufacturer produces 7 models, each available in 6 different colors. In addition, the buyer can choose one of 4 different upholstery fabrics and one of 5 different colors for the interior. How many varieties of cars can be ordered from the manufacturer?
11. How many different telephone numbers can be formed from (a) two different letters and five digits if the first digit cannot be 0; (b) seven digits if the third cannot be O?
12. In long-distance direct dialing, the area code consists of three digits and the local number of seven digits. Taking the area code into consideration, how many telephone numbers can be formed if the sixth digit cannot be O?
13. Each row of a four-rowed signal device contains a red and a green light. If at most one light can be lit in any one row, how many different signals can be sent by this device?
14. A witness to a holdup reports that the license of the getaway car consisted of 6 different digits. He remembers the first three but has forgotten the remainder. How many licenses do the police have to check?
15. In how many ways can you write four-digit numerals, using the digits 3, 4, 5, 6, and 7, if you may use a digit as many times as desired in any one numeral?
16. How many three-letter arrangements can be made of the letters P, R, I, M, and E, if any letter may be repeated?
17. How many positive odd integers whose numerals contain three digits can be formed, using the digits 1,2,3,4, and 5? (Hint: Fill the units place with an odd digit; then fill the remaining places.)
18. How many positive even integers of three digits can be formed from the digits 1,2,3,4, 5, and 6? (Hint: Fill the units place with an even digit; then fill the remaining places.)
19. How many positive odd integers less than 70,000 can be represented, using the digits 2,3,4, 5, and 6?
20. How many positive integers < 8000 can be represented, using 3, 5, 6, 7?
21. How many auto license plates of four symbols can be made in which at least two of the symbols are letters and the rest are digits?
22. How many three-letter code words can be made if at least one of the letters must be one of the vowels: A, E, I, 0, or U?

Section 2: permutations

1. In how many ways can the letters in the word PHOENIX be arranged if each letter is used only once in each arrangement?
2. Seven salesmen are to be assigned to seven different counters in a department store. How many ways can the assignment be made?
3. A school has six sections of first-year algebra. In how many ways can a pair of twins be assigned to algebra classes if their parents have requested that they be placed in different classes?
4. A symphony concert is to consist of five works: 2 modern, 2 of the classical period, and a piano concerto for soloist and orchestra. If the concerto is the last work on the program, how many ways can the program be arranged?
5. How many 4-letter radio station call letters can be made if the first letter must be a W or a K and no letter may be repeated?
6. A beauty salon has 4 assistants who wash hair, 2 stylists who cut hair, and a manicurist. By how many different arrangements of the personnel in the salon may a woman have her hair washed, cut, and then her nails manicured?
7. A business school gives courses in typing, shorthand, transcription, business English, technical writing, and accounting. In how many ways may a student arrange his program if he has three courses a day?
8. A sandlot baseball team has 13 players. If there are three pitchers, two catchers, two center-fielders, and one of each other position, how many 9-man starting teams can be fielded?
9. How many permutations of the letters A N S W E R end in a vowel?
10. How many permutations of the letters E Q U I N 0 X end in a consonant?
11. How many permutations of the letters M O D E R N have consonants in the second and third positions?
12. How many even integers whose numerals contain three digits can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated in a numeral?
13. How many odd natural numbers are there having a 4-digit numeral in which no digit is repeated?
14. How many numbers divisible by 5 can be formed from the digits 1,2, 3, 4, 5, 6, using each digit exactly once in each numeral?
15. To lead a certain cheer, the seven cheerleaders form a circle, each facing the center. In how many orders can they arrange themselves?
16. A milliner wants to arrange six different flowers around the brim of a hat. In how many orders can she place them?
17. In how many ways can five keys be arranged on a key ring?
18. In how many ways can a girl arrange nine charms on a bracelet?
19. In how many ways can five men and five girls be seated at a round table so that each girl is between two men?
20. In how many ways can four young couples be seated as a round table so that girls and boys are seated alternately?

21. Show that 6P4 = 6(5P3). 23. Show that 5Pr = 5(4Pr-1)
22. Show that nP4 = n(_n-1P₃) 24. Show that 5P3 - 5P2 = 2(5P2)

Section 3: counting subsets: combinations

1. How many combinations can be formed from the letters D R E A M, taking two at a time? Show the combinations.
2. How many combinations can be formed from the letters N I L E, taking two at a time? Write the combinations.
3. How many straight lines can be formed by joining any two of five points, no three of which are in a straight line?
4. Seven points lie on the circumference of a circle. How many chords can be drawn joining them?
5. In how many ways can a student choose to answer five questions out of eight on an examination, if the order of his answers is of no importance?
6. Eva's Hamburger Haven sells hamburgers with cheese, lettuce, tomato; relish, ketchup, or mustard. How many different hamburgers can be made, choosing any three of the "extras"?
7. Julie owes letters to her grandmother, her uncle, her cousin, and two school friends. Tuesday night she decides to write to two of them. From how many combinations can she choose?
8. Local 352 is holding an election of four officers. In how many ways can they be chosen from a membership of 75? (Disregard order.)
9. A quality control engineer has to inspect a sample of 5 fuses from a box of 100. How many different samples can he choose?
10. The twelve engineers in the instrumentation department of Ajax Missile Corp. are to divide into project groups of four persons each. How many possible groups are there?
11. Six points lie on the circumference or a circle. How many inscribed triangles can be drawn having these points as vertices?
12. Eleven points lie on the circumference of a circle. How many inscribed hexagons can be drawn having these points as vertices?
13. If 6 students are to be chosen from 12 to participate in an honors section of a course, in how, many ways can this group be selected? In how many ways can the group not chosen to participate be selected?
14. How many ways can a chemistry student choose 5 experiments to perform out of 10? In how many ways can 10 things be divided into 2 equal groups?
15. How many different five-card hands can be drawn from a pack of 52 cards? (Set up solution and estimate answer.)
16. How many different thirteen-card hands can be dealt from the 52 cards in a pack? (Set up solution and estimate answer.)
17. Find n, given nC2 = 100C98.
18. Find n if nC5 = nC3.
19. Prove nCr = nC(n-r) by using the formula nCr = n! / [(n - r)!r!]
20. Show that the total number of subsets of a set with n elements is 2ⁿ. Hint. Each member of the set either is or is not selected in forming a subset.

Section 4: combination and permutation applications

1. Mrs. Henry McGrath has five hats, nine winter dresses, three handbags, and six pairs of shoes. How many different winter outfits does she have?
2. On a geometry test, each student may select one theorem to prove from three which are given and two constructions to perform from three which are given. In how many ways can a student make his selection?
3. Students in an English class are to write a report on two books read outside of class. For the first, they may choose from four different books, and for the second, from three different books. How many different choices of reports are there? (Disregard order.)
4. There are ten men and six women in a repertory theatre group. Four of the men can play male leads, and the others play supporting roles. Three of the women play female leads and the others play supporting roles. In how many ways can a play with male and female leads and two male and three female supporting roles be cast?
5. A chef interested in using up some leftover meats and vegetables decides to make a stew consisting of three kinds of meat and four vegetables. If there are five different meats and seven different vegetables available, how many different kinds of stew can the chef make?
6. Seven boys and seven girls were nominated for homecoming king and queen. How many ways can a king, a queen, and her court of two girls be chosen?
7. A bridge deck has thirteen cards of each suit. How many 13-card hands having seven spades are there? How many hands having exactly seven spades, three hearts, two diamonds, and one club are there?
8. A deck of cards has thirteen cards of each suit. How many 10-card hands having three cards of one kind and two of another are there? How many hands having three 10s and two queens are there?
9. A department-store window designer has eight spring dresses, four shorts-and-shirt outfits, and seven bathing suits from which to choose three dresses, two shorts-and-shirt outfits, and two bathing suits. How many ways can she arrange these on seven manikins?
10. The executive division of a business organization proposed three speakers for the annual banquet, the sales department proposed two, the production department proposed two, and the accounting department proposed one, If there is to be one speaker from each department, how many ways can they be arranged on one side of the head table?
11. How many five-letter arrangements of the letters R E G I 0 N A L consisting of three consonants and two vowels can be formed if no letter is repeated?

SOLUTIONS

Section 1: 7. 676 8. 27 9. 1024 10. 840 11. a. 5.85 x 10⁷ b. 9 x 106 12. 9 x 10⁹ 13. 80
14. 210 15. 625 16. 125 17. 75 18. 108 19. 1562 20. 340 21. 1,565,616 22. 8315

Section 2: 1. 5040       2. 5040      3. 30       4. 24      5. 27,600       6. 8      7. 120       8. 12      9. 240
10.2160      11. 288       12. 24       13. 2240      14. 120       15. 720      16. 120       17. 12      18. 20,160
19. 2880      20. 144

Section 3: 1. 10      2. 6       3. 10      4. 21       5. 56      6. 20       7. 10      8. 1,215,450       9. 75,287,520
10. 495      11. 20      12. 462        13. 924; 924      14. 252; 126      15. approx. 2.6 x 10⁶
16. approx. 7 x 10¹¹      17. n = 100       18. n = 8

Section 4: 1. 810 2. 9 3. 12 4. 180 5. 350 6. 735 7. approx. 5.6 x 10⁹; approx. 5 x 10⁸
8. 4.1 x 10⁹; 2.6 x 10⁷ 9. approx. 3.6 x 10⁷ 10. 288 11. 2880