Topic: Fundamental Principle of Counting, Combination and Permutation Permutation – arrangement, schedule, order Combination – group, committee, sample, selection Examples: 1. A restaurant offers a choice of 3 salad, 5 main dishes, and 2 desserts. Find the number of a different 3 course meals that can be selected. 3x5x2 = 30 2. A teacher has 5 different books that he wishes to arrange in a row. How many different arrangements are possible? 5x4x3x2x1 = 120 3. If the teacher in number 2 wishes to place only 3 books of the 5books in a row, how many arrangement of 3 books are possible? 5x4x3 = 60 4. How many positive integers of 2 different digits can be formed from the integers 1,2,3 and 4? 4x3=12 5. Determine how many ways a chairman and a secretary can be chosen from a committee of 5 people? 5x4 =20 6. If there are 7 doors providing access to a building, in how many ways can a person enter the building by one door and leave by a different door? 7x6 = 42 7. How many different arrangements of 6 distinct books each can be made on a shelf with space for 6 books? 6! = 720 8. How many different arrangements, each consisting of 4 different letters can be formed from the letters of the word “personal” if each arrangement is to begin and end with a vowel? 3x6x5x2=180 9. There are 12 questions on a true or false test. If all questions answered, in how many different ways can the test be completed? 212 = 4096 10. Each of the 8 questions has 3 possible answers. If all questions are answered, in how many ways can the test be completed? 38=6561 11. A quiz consist s of 10 true or false questions. If a student guesses all of the answers, in how many different can she complete the quiz? 210 = 1024 12. A company is setting up phone numbers for a new town. Each number must have 7 digits and must not start with zero. How many different numbers are possible? (Repetition of digits is allowed) 9x106 = 9,000,000 13. A seven-digit phone number is to be chosen so that the first and the last digits are not zero and the last digit is even. 9x10x10x10x10x10x4 = 3,600,000 14. How many different six-digit license plates can be made if the first digit must not be zero, and no digits may be repeated? 9x9x8x7x6x5 = 136,080 15. How many different six- digit license plates can be made if the first digit must not be zero, but the digits may be repeated? 9x10x10x10x10x10 = 900,000 16. Five roads link towns A and B. Three roads link towns B and C. How many roads link towns A and C? 5 x 3 = 15 17. Each of 2 fencing clubs has 100 members and each selects 1 member to represent it at a fencing meet. In how many ways can the2 men be chosen? 100x100 = 10,000 18. There are 5 kinds of envelopes and 4 kinds of stamps of the same denomination. In how many ways can we choose an envelope and a stamp for sending a letter? 5 x 4 = 20 19. In how many ways can we choose a consonant and vowel in the word “paste”? 3 x 2 = 6 20. We cast a 6-sided die and spin an 8-sided top. In how many ways can the two fall? 6 x 8 = 48 21. Five roads lead to the top of a mountain. In how many ways can a tourist go up and down the mountain? What is the answer if the tourist must not retrace his steps? 25; 20 22. There are 20 sheep and 24 pigs on a farm. In how many ways can we choose a sheep and a pig? Having chosen a sheep and a pig, in how many ways can we choose another sheep and another pig? 480; 437 23. In how many ways can we choose 3 words, one each from 12 three-letter words, 9 four-letter words, and fiveletter words? 12x9x10 = 1080 24. Given 6 pairs of gloves of different sizes, in how many ways can we choose 2 gloves of different sizes? 6 x 5 = 30 25. In how many ways can we choose 3 textbooks, one each from 3 copies of an algebra textbook, 7 copies of a geometry textbook, and 7 copies of a trigonometry textbook? 3x7x7=147 26. In how many ways can we choose 3 out of 5 different colors? 5C3 = 10 27. One person has 7 mathematics books and the other has 9 mathematics books. In how many ways can the two people exchange a mathematics book? 7x9 = 63
28. How many 4-digt numbers (with repetitions) can be composed of the digits 0,1,2,3,4,5,6? (No number can start with the digit 0.) 6x7x7x7 = 2058 29. In an expression on social interaction, 6 people will sit on 6 seats in a row. In how many ways can this be done? 30. In how many ways can 7 of 10 monkeys be arranged in a row for a genetic experiment? 31. A business school offers courses in typing, shorthand, transcription, Business English, technical writing, and accounting. In how many ways can a student arrange a schedule if 3 courses are taken? 32. If your college offers 400 courses, 20 of which are in mathematics and your counsellor arranges your schedule of 4 courses by random selection, how many schedules are possible that do not include a Math course? 33. In a club with 5 members, how many ways can a slate of 3 officers consisting of President, Vice-President and Secretary by chosen? 34. A baseball team has 20 players. How many nine player batting orders are possible? 35. In how many ways can 5 players be arranged to the 5 positions on a basketball team, assuming that any player can play any position? In how many ways can 10 players be assigned to the five positions? 36. How many ways can all the letters of the word “TOUGH” be arranged? 37. A club has 30 members. If a committee of 4 is selected at random, how many committees are possible? 38. How many different samples of 3 apples can be drawn from a crate of 25 apples? 39. Hal’s Hamburger Heaven sells hamburger with cheese, relish, lettuce, tomato, mustard or ketchup. How many different hamburgers can be made using any 3 of the extras? 40. Three students are to be selected from a group off 12 students to participate in a special class. In how many ways can this be done? In how many ways can the group that will not participate be selected? 41. Five cards are marked with the numbers 1,2,3,4,5 shuffled, and 2 cards are then drawn. How many different two-card combinations are possible? 42. If a bag contains 15 marbles, how many samples of two marbles can be drawn from it? How many samples of 4 marbles? 43. In #42, if the bag contains 3 yellow, 4 white and 8 blue marbles , how many samples of 2 can be drawn in which both marbles are blue? 44. In a crate of 25 apples, if it is known that there are 5 rotten apples in the crate: a. How many samples of 3 could be drawn in which all 3 are rotten? b. How many samples of 3 could be drawn in which there is 1 rotten apple and 2 good apples? 45. A city council is composed of 5 liberals and 4 conservatives. Three members are to be selected randomly as delegates to a convention. a. How many delegations are possible? b. How many delegations could have all liberals? c. How many delegations could have 2 liberals and 1 conservative? 46. How many ways can 6 different books be placed in a bookshelf? 47. How many ways can 4 books be chosen and arranged in 4 spaces in a bookshelf from 10 different books? 48. How many 2-digit numbers can be formed from the digits 1,2,3,4 if no digits is repeated? If repetition is allowed? 49. In how many ways can 8 people be seated at a table? 50. In how many ways can 11 people be seated at a table? 51. How many distinct permutations can be formed from the letters of the “STATISTICS”? 52. In how many ways can 4 boys and 3 girls be seated in a row of 5 chairs? 53. In how many ways can 3 of 10 students participating in a international contest be rank first, second and third? 54. In how many ways may a coach select his first five basketball players from his 12-member team? 12P5 = 792 55. If there will be 20 contenders for the 12 seats in 2010 senate race, how many ways may a voter select his 12 senators? 20C12 = 125,970 56. If each letter of the word “EXCELLENCE” will be rearranged to form a new word with a new meaning. How many new words will there be? (10!)/(4!x2!x2!) = 37800 57. In how many ways may 9 distinct colored beads be arranged? In how many ways may the beads form a bracelet? 9! ; 8! 58. In how many ways can 4 boys and 4 girls be seated in a row if a. They may sit anywhere? 8! b. The boys and girls must alternate? (4!4!)2 c. The girls must be together? (girls = 1 group + 4 boys) = 5 groups 5!4! = 2880 d. A particular boy B and a particular girl G cannot sit beside each other? 8! – 7!2! (7 groups)
59. Ten points are taken on a circumference of a circle. a. How many chords can be drawn by joining them in all possible ways? 10C2 b. How many triangles can be drawn having these points as vertices? 10C3 c. How many hexagons can be drawn? 10C6 60. How many alphanumeric car license plate numbers are there? How many plate numbers are there if only the letters P,N,T,X,andZ will be used without repetition of the numbers? a. 10x10x10x26x20x26 ; b. 10x 9x8x5x5x5 61. How many three-digiteven numbers are there using the numbers 1,2,3,4,5,6 without repetition? With repetition? 6x5x3 ; 6x6x3 62. How many rearrangements of the word REDEEMER are there? 840 63. Five scholars will be chosen from 12 honor students in a class. How many ways may the scholar be chosen? 12C5 64. How many ordered pairs using 7 natural numbers and the letters: a,b,c,d,e may be formed? 35 65. A college student has 2 maong pants, 4 polo shirts and 2 pairs of shoes to wear. How many ways can he form a distinct outfit? 2x4x2 = 16 66. If the Senate Committee must be composed of 6 from Liberal Party and 6 from Nacionalista Party, in how many ways may 9 and 12 contenders from the two parties respectively, be chosen? 9C6 x 12C6 67. If a class of 30 students will form groups of 6 students, how many possible groups will there be? 30C6 68. In how many ways may 10 distinct colored gem stones be arranged? In how many ways may the beads form a necklace? 69. How many permutations of the letter in the word TIDE are there? 24 70. In how many ways can a committee of 4 be formed from a group of 12? 12C4 = 495 71. Six points lie on a plane, no three of which are collinear. In how many ways can a person draw a. A line? 6C2 b. A pentagon? 6C5 c. A quadrilateral? 6C4 d. A hexagon? 6C6 72. How many handshakes can be made by a. 10 people? 10C2 b. 5 people? 5C2 73. A catering service offers 5 appetizers, 8 main courses, and 5 desserts. A banquet committee is to select 3 appetizers, 4 main courses and 2 desserts. How many ways can this be done? 5C3 x 8C4 x 5C2 = 7000 74. In how many ways can you arrange the letters in the word a. BARBIE? 360 b. DEER? 12 75. A group off 7 select a president, a vice-president, a treasurer and a secretary. How many ways can this be done? 7P4 = 840 76. If Matthew has 8 courses to choose from, how many ways can he arrange his schedule, if he must pick 4 off them? 8P4 = 1680 77. How many samples of 3 pineapples can be drawn from a crate of 12? 12C3 78. In a club with 8 men and 11 women members, how many 5-member committees, can be chosen that have the following: a. All men? 8C5 b. All women? 11C5 c. 3 men and 2 women? 8C3 x 11C2 d. At least 3 women? e. At most 3 women? 79. How many two-digit numbers may be formed from each given sets of digits a. If repetitions are not allowed? b. If repetitions are allowed? 79.1 { 0, 1,2,3} a. 3x3 = 9 b. 3x 4 = 12 79.2 { 0, 1, 2, 3, 4, 5} a. 5x5 = 25 b. 5 x 6 = 30 80. How many three-digit numbers can be formed from the set { 1,2,3,…,9} if a. repetitions are not allowed? b. repetitions are allowed? 80.1 even? a. 8x7x4 = 224 b. 8x8x4 = 256
80.2 divisible by 5? a. 8x7x1 = 56 b. 8x8x1=64 81. A model selecting her outfits purchased 5 blouses, 4 skirts and 3 blazers. How many different new outfits new outfits consisting of a blouse, a skirt and a blazer can she create from her new collection? 5x4x3 = 60 82. Find the number of different arrangements of the set of six letters {h, o, n, e, s,t} a. taken two at a time. 6P2 = 30 b. taken 6 at a time. 6P6 = 720 83. In how many different orders can the letters in the set {a, e, i, o , u } be written? 5! 84. Find the number of permutations of the letters of each word. a. BALL 2 b. BIBLE 60 85. In how many different ways can 9 people be seated at a round table? 8! 86. How many different numerals can be formed using all the given numeral? a. 622581 6!/2! = 360 b. 3334777 7!/(3!3!) = 140 87. How many combinations of 5 records can be chosen from 12 records 12 records offered by a record club? 12C5 = 792 88. How many lines are determined by 8 points, no three points of which are collinear? 8C2 = 28 89. A class consists of 12 boys and 15 girls. How many different committees of 4 can be selected from the class if each committee is to consist of 2 boys and 2 girls? 12C2 x 15C2 = 66x105 = 6,930 90. There are 12 people in a room. If each of them shake hands with each other, how many handshakes are there? 12C2 = 66 91. A box of 10 articles contains one that is defective. In how many ways as Marivic select 4 articles such that a. the defective article is included? 9C3 x 1C1 = 84 b. the defective article is not included? 9C4 = 126 92. A box contains 7 red balls, 5 white and 3 green balls. In how many ways can we select 3 balls such that a. they are all red? 7C3 = 35 b. they are all white? 5C3 =10 c. they are all green? 3C3 = 1 d. they are of different colors? 7 x 5x 3 = 105 e. two are red and one is white? 7C2 x 5C1 = 21 x 5 = 105 f. exactly two are white? 5C2 x 10C1 = 100 g. exactly 1 is green? 3C1 x12C2 = 3 x 66 = 198 h. none is green? 12 C3 = 220 93. How many ways can 4 people sit at a round table? 3! = 6 94. How many ways can 8 people sit at a round a table? 7! = 5040 95. How many ways can 7 keys be arranged on a key ring? 6!/2 = 360 96. How many ways can 3 men and 3 women sit at a round table? 5! = 120 97. How many ways can seven different books be placed in a book shelf? 7! 98. How many ways can 5 books be chosen and arranged in 5 spaces in a bookshelf from 10 different books? 10P5 = 30,240 99. In how many ways can 6 people be seated at a round table? 5! 100. How many distinct permutations can be formed from the letters of the word “error”? 5!/3! = 20 101. In how many ways can 5 of 10 students participating in a n international contest to rank first, second, third, fourth, and fifth? 10P5 = 30240 102. In how many ways can a committee of 5 be chosen from a group of 7 people? 7C5 =21 103. In how many ways can we select 3 spades and 4 diamonds from a deck of 52 cards? 13C3 x13C4 = (286)(715)= 204,490 104. A club consists of 10 boys and 15 girls. They wish to elect officers consisting of a boy as president and a girl as vicepresident. They also wish to elect a treasurer and a secretary, who may be of either sex. How many sets of officers are possible? 10x15x23x22 = 75900 105. How many four-digit whole numbers can be formed if repetitions of digits are allowed? How many if repetitions are not allowed? a. 9x10x10x10 = 9000 b. 9x9x8x7=4536
Topic: Fundamental Principle of Counting, Combination and Permutation Permutation – arrangement, schedule, order Combination – group, committee, sample, selection Examples: 1. A restaurant offers a choice of 3 salad, 5 main dishes, and 2 desserts. Find the number of a different 3 course meals that can be selected. 2. A teacher has 5 different books that he wishes to arrange in a row. How many different arrangements are possible? 3. If the teacher in number 2 wishes to place only 3 books of the 5books in a row, how many arrangement of 3 books are possible? 4. How many positive integers of 2 different digits can be formed from the integers 1,2,3 and 4? 5. Determine how many ways a chairman and a secretary can be chosen from a committee of 5 people? 6. If there are 7 doors providing access to a building, in how many ways can a person enter the building by one door and leave by a different door? 7. How many different arrangements of 6 distinct books each can be made on a shelf with space for 6 books? 8. How many different arrangements, each consisting of 4 different letters can be formed from the letters of the word “personal” if each arrangement is to begin and end with a vowel? 9. There are 12 questions on a true or false test. If all questions answered, in how many different ways can the test be completed? 10. Each of the 8 questions has 3 possible answers. If all questions are answered, in how many ways can the test be completed? 11. A quiz consist s of 10 true or false questions. If a student guesses all of the answers, in how many different can she complete the quiz? 12. A company is setting up phone numbers for a new town. Each number must have 7 digits and must not start with zero. How many different numbers are possible? (Repetition of digits is allowed) 13. A seven-digit phone number is to be chosen so that the first and the last digits are not zero and the last digit is even. 14. How many different six-digit license plates can be made if the first digit must not be zero, and no digits may be repeated? 15. How many different six- digit license plates can be made if the first digit must not be zero, but the digits may be repeated? 16. Five roads link towns A and B. Three roads link towns B and C. How many roads link towns A and C? 17. Each of 2 fencing clubs has 100 members and each selects 1 member to represent it at a fencing meet. In how many ways can the2 men be chosen? 18. There are 5 kinds of envelopes and 4 kinds of stamps of the same denomination. In how many ways can we choose an envelope and a stamp for sending a letter? 19. In how many ways can we choose a consonant and vowel in the word “paste”? 20. We cast a 6-sided die and spin an 8-sided top. In how many ways can the two fall? 21. Five roads lead to the top of a mountain. In how many ways can a tourist go up and down the mountain? What is the answer if the tourist must not retrace his steps? 22. There are 20 sheep and 24 pigs on a farm. In how many ways can we choose a sheep and a pig? Having chosen a sheep and a pig, in how many ways can we choose another sheep and another pig? 23. In how many ways can we choose 3 words, one each from 12 three-letter words, 9 four-letter words, and fiveletter words? 24. Given 6 pairs of gloves of different sizes, in how many ways can we choose 2 gloves of different sizes? 25. In how many ways can we choose 3 textbooks, one each from 3 copies of an algebra textbook, 7 copies of a geometry textbook, and 7 copies of a trigonometry textbook? 26. In how many ways can we choose 3 out of 5 different colors? 27. One person has 7 mathematics books and the other has 9 mathematics books. In how many ways can the two people exchange a mathematics book? 28. How many 4-digt numbers (with repetitions) can be composed of the digits 0,1,2,3,4,5,6? (No number can start with the digit 0.)
29. In an expression on social interaction, 6 people will sit on 6 seats in a row. In how many ways can this be done? 30. In how many ways can 7 of 10 monkeys be arranged in a row for a genetic experiment? 31. A business school offers courses in typing, shorthand, transcription, Business English, technical writing, and accounting. In how many ways can a student arrange a schedule if 3 courses are taken? 32. If your college offers 400 courses, 20 of which are in mathematics and your counsellor arranges your schedule of 4 courses by random selection, how many schedules are possible that do not include a Math course? 33. In a club with 5 members, how many ways can a slate of 3 officers consisting of President, Vice-President and Secretary by chosen? 34. A baseball team has 20 players. How many nine player batting orders are possible? 35. In how many ways can 5 players be arranged to the 5 positions on a basketball team, assuming that any player can play any position? In how many ways can 10 players be assigned to the five positions? 36. How many ways can all the letters of the word “TOUGH” be arranged? 37. A club has 30 members. If a committee of 4 is selected at random, how many committees are possible? 38. How many different samples of 3 apples can be drawn from a crate of 25 apples? 39. Hal’s Hamburger Heaven sells hamburger with cheese, relish, lettuce, tomato, mustard or ketchup. How many different hamburgers can be made using any 3 of the extras? 40. Three students are to be selected from a group off 12 students to participate in a special class. In how many ways can this be done? In how many ways can the group that will not participate be selected? 41. Five cards are marked with the numbers 1,2,3,4,5 shuffled, and 2 cards are then drawn. How many different two-card combinations are possible? 42. If a bag contains 15 marbles, how many samples of two marbles can be drawn from it? How many samples of 4 marbles? 43. In #42, if the bag contains 3 yellow, 4 white and 8 blue marbles , how many samples of 2 can be drawn in which both marbles are blue? 44. In a crate of 25 apples, if it is known that there are 5 rotten apples in the crate: a. How many samples of 3 could be drawn in which all 3 are rotten? b. How many samples of 3 could be drawn in which there is 1 rotten apple and 2 good apples? 45. A city council is composed of 5 liberals and 4 conservatives. Three members are to be selected randomly as delegates to a convention. a. How many delegations are possible? b. How many delegations could have all liberals? c. How many delegations could have 2 liberals and 1 conservative? 46. How many ways can 6 different books be placed in a bookshelf? 47. How many ways can 4 books be chosen and arranged in 4 spaces in a bookshelf from 10 different books? 48. How many 2-digit numbers can be formed from the digits 1,2,3,4 if no digits is repeated? If repetition is allowed? 49. In how many ways can 8 people be seated at a table? 50. In how many ways can 11 people be seated at a table? 51. How many distinct permutations can be formed from the letters of the “STATISTICS”? 52. In how many ways can 4 boys and 3 girls be seated in a row of 5 chairs? 53. In how many ways can 3 of 10 students participating in a international contest be rank first, second and third? 54. In how many ways may a coach select his first five basketball players from his 12-member team? 55. If there will be 20 contenders for the 12 seats in 2010 senate race, how many ways may a voter select his 12 senators? 56. If each letter of the word “EXCELLENCE” will be rearranged to form a new word with a new meaning. How many new words will there be? 57. In how many ways may 9 distinct colored beads be arranged? In how many ways may the beads form a bracelet? 58. In how many ways can 4 boys and 4 girls be seated in a row if a. They may sit anywhere? b. The boys and girls must alternate? c. The girls must be together? d. A particular boy B and a particular girl G cannot sit beside each other? 59. Ten points are taken on a circumference of a circle. a. How many chords can be drawn by joining them in all possible ways? b. How many triangles can be drawn having these points as vertices?
c. How many hexagons can be drawn? 60. How many alphanumeric car license plate numbers are there? How many plate numbers are there if only the letters P,N,T,X,andZ will be used without repetition of the numbers? 61. How many three-digiteven numbers are there using the numbers 1,2,3,4,5,6 without repetition? With repetition? 62. How many rearrangements of the word REDEEMER are there? 63. Five scholars will be chosen from 12 honor students in a class. How many ways may the scholar be chosen? 64. How many ordered pairs using 7 natural numbers and the letters: a,b,c,d,e may be formed? 65. A college student has 2 maong pants, 4 polo shirts and 2 pairs of shoes to wear. How many ways can he form a distinct outfit? 66. If the Senate Committee must be composed of 6 from Liberal Party and 6 from Nacionalista Party, in how many ways may 9 and 12 contenders from the two parties respectively, be chosen? 67. If a class of 30 students will form groups of 6 students, how many possible groups will there be? 68. In how many ways may 10 distinct colored gem stones be arranged? In how many ways may the beads form a necklace? 69. How many permutations of the letter in the word TIDE are there? 70. In how many ways can a committee of 4 be formed from a group of 12? 71. Six points lie on a plane, no three of which are collinear. In how many ways can a person draw a. A line? b. A pentagon? c. A quadrilateral? d. A hexagon? 72. How many handshakes can be made by a. 10 people? b. 5 people? 73. A catering service offers 5 appetizers, 8 main courses, and 5 desserts. A banquet committee is to select 3 appetizers, 4 main courses and 2 desserts. How many ways can this be done? 74. In how many ways can you arrange the letters in the word a. BARBIE? b. DEER? 75. A group off 7 select a president, a vice-president, a treasurer and a secretary. How many ways can this be done? 76. If Matthew has 8 courses to choose from, how many ways can he arrange his schedule, if he must pick 4 off them? 77. How many samples of 3 pineapples can be drawn from a crate of 12? 78. In a club with 8 men and 11 women members, how many 5-member committees, can be chosen that have the following: a. All men? b. All women? c. 3 men and 2 women? d. At least 3 women? e. At most 3 women? 79. How many two-digit numbers may be formed from each given sets of digits a. If repetitions are not allowed? b. If repetitions are allowed? 80. How many three-digit numbers can be formed from the set { 1,2,3,…,9} if a. repetitions are not allowed? b. repetitions are allowed? 80.1 even? 80.2 divisible by 5? 81. A model selecting her outfits purchased 5 blouses, 4 skirts and 3 blazers. How many different new outfits new outfits consisting of a blouse, a skirt and a blazer can she create from her new collection? 82. Find the number of different arrangements of the set of six letters {h, o, n, e, s,t} a. taken two at a time. b. taken 6 at a time. 83. In how many different orders can the letters in the set {a, e, i, o , u } be written? 84. Find the number of permutations of the letters of each word. a. BALL b. BIBLE
85. In how many different ways can 9 people be seated at a round table? 86. How many different numerals can be formed using all the given numeral? a. 622581 b. 3334777 87. How many combinations of 5 records can be chosen from 12 records 12 records offered by a record club? 88. How many lines are determined by 8 points, no three points of which are collinear? 89. A class consists of 12 boys and 15 girls. How many different committees of 4 can be selected from the class if each committee is to consist of 2 boys and 2 girls? 90. There are 12 people in a room. If each of them shake hands with each other, how many handshakes are there? 91. A box of 10 articles contains one that is defective. In how many ways as Marivic select 4 articles such that a. the defective article is included? b. the defective article is not included? 92. A box contains 7 red balls, 5 white and 3 green balls. In how many ways can we select 3 balls such that a. they are all red? b. they are all white? c. they are all green? d. they are of different colors? e. two are red and one is white? f. exactly two are white? g. exactly 1 is green? h. none is green? 93. How many ways can 4 people sit at a round table? 94. How many ways can 8 people sit at a round a table? 95. How many ways can 7 keys be arranged on a key ring? 96. How many ways can 3 men and 3 women sit at a round table? 97. How many ways can seven different books be placed in a book shelf? 98. How many ways can 5 books be chosen and arranged in 5 spaces in a bookshelf from 10 different books? 99. In how many ways can 6 people be seated at a round table? 100. How many distinct permutations can be formed from the letters of the word “error”? 101. In how many ways can 5 of 10 students participating in a n international contest to rank first, second, third, fourth, and fifth? 102. In how many ways can a committee of 5 be chosen from a group of 7 people? 103. In how many ways can we select 3 spades and 4 diamonds from a deck of 52 cards? 104. A club consists of 10 boys and 15 girls. They wish to elect officers consisting of a boy as president and a girl as vicepresident. They also wish to elect a treasurer and a secretary, who may be of either sex. How many sets of officers are possible? 105. How many four-digit whole numbers can be formed if repetitions of digits are allowed? How many if repetitions are not allowed?
