MAT104: Fundamentals of Mathematics II Counting Techniques Class Exercises Solutions 1. Appetizers: Salads: Entrées: Desserts: 2. Letters: (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z) Colors: (Red, Orange, Yellow, Green, Blue, Indigo, Violet) Digits: Months: (January, February, March, April, May, June, July, August, September, October, November, December) Suits: (Clubs, Diamonds, Hearts, Spades) 3. Months: Days: 4. Die: Card: 5. Heart: Face: Both:
6. Even: total ways to roll even Multiple of : ways to roll multiple of Both even and multiple of : ways to roll even multiple of 7. Full time work: Attend college: Work and college: 8. Total: Math: Neither: Science: Neither: Math or science: Math and science: Math: Science: Math and science:
9. Digits allowed: Divisors of in list of digits allowed: First Second Third Final Digit Digit Digit Number Using tree diagram above, number of three-digit numbers matching specified criteria:
10. 11. 12. Ending with zero: Not ending with zero: Total: 13. 14. ending with : ending with : : : Total:
15. Red: White: Blue: Green: Total: 16. Each card can be assigned to any one of the three students. 17. Denote the three students as students A, B, and C. 18. 19. Number of students: Number of possible grades per student: 20. Number of letters in one word: Number of possibilities for each letter: 21. Number of questions: Number of possible answers per question:
22. Number of questions: Number of possible answers per question: 23. 24. 25. (actual answer: ) 26. Number of tigers: Number of elephants: Number of dogs: Number of species: 27. Number of freshmen: Number of sophomores: Number of juniors: Number of seniors: Number of groups: 28.
29. 30. 31.
32. 33. Black socks: Blue socks: Green socks: Total socks: A pair consists of any two socks in the drawer if colors do not have to match.
34. Black socks: Blue socks: Green socks: A pair consists of two socks of the same color and can be black, blue, or green. Black pairs: Blue pairs: Green pairs: Total pairs:
35. 36. rats: (from Exercise #35 above) rats: rats: At least rats:
37. If Snow White must serve on the committee, then of the Seven Dwarves must serve. Snow White: of the Seven Dwarves: Snow White and of the Seven Dwarves:
38. Male students: Female students: Total students: 39. boys: girls: boys and girls:
40. white hats: blue hats: white and blue hats:
41. Total apples: Rotten apples: Good apples: A sample of apples having rotten apples implies the sample also contains good apples. rotten apples: good apples: rotten apples and good apples: From this point forward, the details of calculations involving combinations and permutations are omitted in the interest of space, although key intermediate results are still included in all solutions.
42. Total iphones: Damaged iphones: Good iphones: No more than damaged iphones implies,, or damaged iphones. A sample of iphones with damaged implies the sample also contains good iphones. A sample of iphones with damaged implies the sample also contains good iphones. A sample of iphones with damaged implies the sample also contains good iphones. damaged and good iphones: damaged and good iphones: damaged and good iphones: No more than damaged iphones: 43. A flush means all cards are of the same suit, which can be thought of as follows: Select of the suits in the deck, and then select of the cards in that suit. Since the order of selection is irrelevant, this exercises lends itself to combinations. Select suit: Select cards from the selected suit: Flushes: 44. Pair of threes: select of the threes in the deck Pair of nines: select of the nines in the deck Face card: select of the face cards in the deck Pair of threes and pair of nines and face card:
45. Two pair means two cards of one denomination, two cards of a different denomination, and one card of yet a different denomination. Since the order in which the two denominations from which to form the two pairs are selected is irrelevant, selecting those two denominations involves combinations. Obtaining two pair can be described as follows: Select denominations from the in the deck from which to form the pairs, and then select of the cards in the first of those denominations to form the first pair, and then select of the cards in the second of those denomination to form the second pair, and then select of the remaining denominations from which to obtain the final card in the hand, and then select of the cards in that denomination to be the final card in the hand. Using that breakdown, the number of hands that are classified as two pair is the following: 46. A poker hand containing exactly spades implies non-spades are in the hand of cards. Obtaining three spades can be described as follows: Select of the spades in the deck, and then select any of the remaining cards in the deck. Using that breakdown, the number of hands containing exactly three spades is the following: 47. Obtaining a full house can be described as follows: Select denomination from the in the deck from which to form three-of-a-kind, and then select of the cards in that denomination to form the three-of-a-kind, and then select of the remaining denominations from which to form a pair, and then select of the cards in that denomination to form the pair. Using that breakdown, the number of hands that are classified as a full house is the following: (continued on next page)
47. (continued) An alternative approach is to realize that a full house formed by having three of one denomination and two of another denomination is different from a full house formed by having two of the first denomination and three of the second denomination. In such a case, the order in which those two denominations are selected matters, so permutations can be used for that portion of the exercise. This approach can be described as follows: Select of the denominations from the in the deck in a specific order from which to form three-of-a-kind and a pair, respectively, and then select of the cards in the first denomination to form the three-of-a-kind, and then select of the cards in the second denomination to form the pair. Using that breakdown, the number of hands that are classified as a full house is the following: 48. 49. At most heads implies,, or heads. heads: head: heads: Number of ways to obtain at most heads: 50. At least heads is the opposite of less than heads, so the complement rule applies. Less than heads implies heads or head. heads: heads: Number of ways to obtain less than heads: Total number of outcomes on Number of ways to obtain at least flips of a fair coin: heads: [ ]
51. Obtaining exactly two vowels by randomly selecting six letters of the English alphabet can be described as follows: Select of the vowels in the English alphabet, and then select of the consonants. Using that breakdown, the number of ways in which exactly two vowels can be selected if six letters of the English alphabet are chosen at random is the following: 52. Number of Ms: Number of Is: Number of Ss: Number of Ps: Total number of letters: 53. Number of As: Number of Bs: Number of Rs: Number of Cs: Number of Ds: Total number of letters:
54. Number of Bs: Number of As: Number of Ns: Number of Cs: Total number of letters: 55. In a worst-case scenario for this situation, the birth months of the individuals in the group are dispersed as much as possible among the calendar months. Consequently, there must be at least one person more than the number of calendar months to guarantee a birth month is repeated among the group members. Minimum number of people to guarantee a birth month is repeated: 56. Possible totals on each roll of a pair of dice: Number of unique totals that can be rolled on a single roll of a pair of dice: In a worst-case scenario for this situation, each roll of the pair of dice produces a unique total that has not yet appeared until all the different totals have occurred. Consequently, there must be at least one roll more than the number of unique totals that can be obtained when rolling a pair of dice to guarantee a total is obtained at least twice. Minimum number of rolls to guarantee a total is repeated:
57. In a worst-case scenario for this situation, the values obtained on each spin are dispersed as evenly as possible across the values shown on the spinner. This means that each value will be obtained once before any value is obtained a second time, each value will be obtained twice before any value is obtained a third time, and so on. As a result, no value will be spun times until each value is obtained times. Consequently, there must be at least one spin more than times the number of values shown on the spinner to guarantee a value is spun times. Number of values on the spinner: Number of spins to obtain occurrences of each value: Minimum number of spins to guarantee a value is obtained at least times: Alternatively, requiring a value is obtained times is equivalent to determining at what point a value is obtained more than times. If the values spun are to be spread out as much as possible among the values shown on the spinner, dividing the number of spins by represents the number of times each value has been obtained at any given point. Letting represent the number of spins needed to accomplish the goal, the following algebraic inequality can be solved: This solution implies that some value must be obtained at least times once the spinner has been spun more than times, which leads to the same result as above.