DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

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$Math 664 Homework #2: Solutions 1. Let Î© = R, F = {A â R : either A ...$

(d) In how many ways can you pair up all the members of the club for singles matches? (e) Suppose that in addition to specifying who plays whom for each pairing you also specify who serves first. Now in how many ways can you specify your pairs? Use your solution to part (d), if possible. 103. Suppose you plan to put six distinct computers in a network as shown in Figure 3.2 where the computers are the nodes. (a) What is the total number of ways to assign computers to the nodes (or the vertices) of this regular hexagon? (b) The edges (line segments) show which computers can communicate directly with which others. Consider two ways of assigning computers to the nodes of the network to be different if there are at least two computers that communicate directly in one assignment and that do not communicate directly in the other. Define an equivalence relation which models the equivalence in this situation. (c) Prove every equivalence class has 48 elements. (d) In how many different ways can you assign computers to the network? 3.3 Counting Subsets Figure 3.2: A computer network. The symbol � � n k is used to represent the number of ways to choose a kelement subset from an n-element set. You may have already seen this notation elsewhere, but do not be concerned if you have not seen it because it will be developed completely here. The symbol should be read as “n choose k”. (Another common way to read the binomial coefficient notation is “the number of combinations of n things taken k at a time” but we’ve found that this can be confusing and so please do not read the notation that 45
way in this course.) Sometimes � � n k is written as C(n, k), or nCk, but � � n k is more standard in discrete mathematics and should be the only notation you use here. •104. Using only the given definition of � � n k (that it equals the number of ways to choose a k-element subset from an n-element set) and the Bijection Principle, prove that � � � � n n = for all 0 ≤ k ≤ n . k n − k 105. Use the distributive law to check that (x + y) 4 = x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 . Explain why this means that � � � � 4 4 = 4 and = 6. (Think this through 1 2 carefully!) Give a general argument for why the coefficient of xkyn−k in (x + y) n equals � � � � n n k . For this reason, the symbol k is called a binomial coefficient. In the next problems you will use equivalence relations to find a formula for calculating � � n k . Although you may have seen this formula before, do not calculate the actual numbers in these problems but rather simply use binomial coefficients in your answers. •106. A basketball team has 12 members of which only five can play at any given time during a game. (a) In how many ways may the coach choose the five players? (b) To be more realistic, the five players normally consist of two guards, two forwards, and one center. If there are five guards, four forwards, and three centers on the team, in how many ways may the coach choose the team to be sent out on the court? (c) If one of the centers is equally skilled at playing forward, how many different teams may the coach send out to play? •107. In how many ways can you pass out k (identical) ping-pong balls to n children if each child may get at most one? (Ask yourself “What is a problem like this doing in the middle of a bunch of problems about counting subsets of a set? Is it related? Or is it supposed to give a break from sets?”) ◦108. Letting S denote the set of all 3-element permutations of {a, b, c, d, e}, Table 3.1 lists all elements of S exactly once and the list is given in 46
Page 1 and 2: DISCRETE MATHEMATICS THROUGH GUIDED
Page 3 and 4: Contents I COURSE NOTES 2 1 Beginni
Page 5 and 6: Part I COURSE NOTES 2
Page 7 and 8: to problems that are particularly i
Page 9 and 10: While working the next problems, be
Page 11 and 12: This idea of using functions to cou
Page 13 and 14: Figure 1.2: The union of four disjo
Page 15 and 16: The domain and co-domain of the fun
Page 17 and 18: •20. If B1, B2, . . . , Bm is a p
Page 19 and 20: In the last problem you used the fa
Page 21 and 22: ◦32. Prove that any function from
Page 23 and 24: nodes) and the line segments are ca
Page 25 and 26: niques which you have used and shou
Page 27 and 28: This illustrates one reason why som
Page 29 and 30: observation is the key to describin
Page 31 and 32: een used above is rather restrictiv
Page 33 and 34: to [10]. Give an inductive process
Page 35 and 36: 66. Are more moves required if the
Page 37 and 38: a recurrence can have infinitely ma
Page 39 and 40: prove your answer is correct, but s
Page 41 and 42: •86. In combinatorics, an (ordere
Page 43 and 44: ◦89. Now again suppose you are ch
Page 45 and 46: 3.2 Equivalence Classes Next is an
Page 47: are in different relative positions
Page 51 and 52: Calculation of Binomial Coefficient
Page 53 and 54: 118. Let S(k, n) denote the number
Page 55 and 56: Notice that a lattice path from (0,
Page 57 and 58: The suggestion intended by the hint
Page 59 and 60: Provided you can show that there is
Page 61 and 62: Chapter 4 Graph Theory 4.1 Undirect
Page 63 and 64: ◦142. Find the degree of each ver
Page 65 and 66: A cycle in a graph is a walk which
Page 67 and 68: and then figure out in how many dif
Page 69 and 70: can begin to see this by working th
Page 71 and 72: 185. Use the definition of the Rams
Page 73 and 74: (b) Draw another connected graph wi
Page 75 and 76: Figure 4.8: A graph. 1 2 4.6 Findin
Page 77 and 78: 202. Use Stirling’s Formula to sh
Page 79 and 80: other restrictions are placed on th
Page 81 and 82: of x n is the number of ways to cho
Page 83 and 84: generating polynomial for the numbe
Page 85 and 86: Express this coefficient in the for
Page 87 and 88: chosen from the singleton set {i} .
Page 89 and 90: coefficient of x k in � ∞� ai
Page 91 and 92: In the last problem you represented
Page 93 and 94: The hardest part of generalizing yo
Page 95 and 96: 248. A more elegant proof can be ob
Page 97 and 98: if and only if f(x) �= i for ever
Page 99 and 100:
•262. Let G be a graph with verte
Page 101 and 102:
Chapter 7 Distribution Problems 7.1
Page 103 and 104:
◦271. In how many ways may you st
Page 105 and 106:
•282. (a) Prove the Bell Numbers
Page 107 and 108:
Index Kn, 20, 58 n!, Stirling’s f
Page 109 and 110:
one-to-one function, 8 onto functio
Page 111 and 112:
Part II REVIEW MATERIAL 108
Page 113 and 114:
Exercise A.2. Here are some questio
Page 115 and 116:
(a) Draw the digraph of the functio
Page 117 and 118:
an n-element set has 2 n subsets. O
Page 119 and 120:
of B1 is 2 k−1 . Consider the fun
Page 121:
Appendix C More on Equivalence Rela
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