738 CHAPTER 11 Probability 46. a) A Dinner Toast Four people at dinner make a toast. If each person is to tap glasses with each other person one at a time, how many taps will take place? 6 b) Repeat part (a) with five people. 10 c) How many taps will there be if there are n people at the dinner table? Cn 2 47. Pascal’s Triangle The notation Cn r may be written n r . ⎛ ⎝⎜ ⎞ ⎠⎟ a) Use this notation to evaluate each of the combinations in the following array. Form a triangle of the results, similar to the one given, by placing the answer to each combination in the same relative position in the triangle. * a0 0b a1 0b a 1 1b a2 0b a 2 1b a 2 2b a3 0b a 3 1b a 3 2b a 3 3b a4 0b a 4 1b a 4 2b a 4 3b a 4 4b b) Using the number pattern in part (a), determine the next row of numbers of the triangle (known as Pascal’s triangle). * 48. Lottery Combinations Determine the number of combinations possible in a state lottery where you must select a) 6 of 46 numbers. 9,366,819 b) 6 of 47 numbers. 10,737,573 c) 6 of 48 numbers. 12,271,512 d) 6 of 49 numbers. 13,983,816 e) Does the number of combinations increase by the same amount going from part (a) to part (b) as from part (b) to part (c)? No 49. a) Table Seating Arrangements How many distinct ways can four people be seated in a row? 24 b) How many distinct ways can four people be seated at a circular table? 24 50. Forming a Committee A group of 15 people wants to form a committee consisting of a chair, vice chair, and three additional members. How many different committees can be formed? 60,060 Recreational Mathematics 51. Card Hands In popular card games, there is such a variety of possible combinations of cards that a player rarely gets the same hand twice. Use the following information about card games to determine the number of different hands. a) Poker: 5 cards are dealt from a standard 52-card deck. 2,598,960 b) Gin Rummy: 10 cards are dealt from a standard 52-card deck. 15,820,024,220 c) Bridge, Hearts, or Spades: 13 cards are dealt from a standard 52-card deck. 635,013,559,600 d) Euchre: 5 cards are dealt from a 24-card deck. 42,504 e) Pinochle: 12 cards are dealt from a 48-card deck. 69,668,534,468 52. Nerts In the card game Nerts, each player starts the game with a standard 52-card deck. To establish the starting position of the game, each player shuffles their cards and then places four cards, face up, in front of themselves. Determine the total number of different starting positions when the number of players is a) four. 1,082,900 b) five. 1,353,625 c) six. 1,624,350 53. a) Combination Lock To open a combination lock, you must know the lock’s three-number sequence in its proper order. Repetition of numbers is permitted. Why is this lock more like a permutation lock than a combination lock? Why is it not a true permutation problem? The order is important. Since the numbers may be repeated, it is not a true permutation lock. b) Assuming that a combination lock has 40 numbers, determine how many different three-number arrangements are possible if repetition of numbers is allowed. 64,000 c) Answer the question in part (b) if repetition is not allowed. 59,280 Research Activity 54. Combinatorics The area of mathematics called combinatorics is the science of counting. Do research and write a paper on combinatorics and its many applications. *See Instructor Answer Appendix
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