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Discrete and Combinatorial Mathematics An Applied Introduction 5th edition Ralph P. Grimaldi - Solutions

Find the coefficient of x15 in each of the following. (a) x3(1 - 2x)10 (b) (x3 - 5x)/(l - x)3 (c) {1 + x)A/{\-x)4
Find all partitions of 7.
Using a Ferrers graph, show that the number of partitions of ft is equal to the number of partitions of 2n into n summands.
Determine the generating function for the sequence a0, a1, a2, ..., where an is the number of partitions of the nonnegative integer ft into (a) even summands; (b) distinct even summands; and (c) distinct odd summands.
What is the generating function for the number of partitions of n ∈ N into summands that (a) cannot occur more than five times; and (b) cannot exceed 12 and cannot occur more than five times?
Show that the number of partitions of a positive integer n where no summand appears more than twice equals the number of partitions of n where no summand is divisible by 3.
Show that the number of partitions of n ∈ Z+ where no summand is divisible by 4 equals the number of partitions of n where no even summand is repeated (although odd summands may or may not be repeated).
Using a Ferrers graph, show that the number of partitions of an integer n into summands not exceeding m is equal to the number of partitions of n into at most m summands.
Find the exponential generating function for each of the following sequences. (a) 1, -1, 1, -1, 1, -1, . . . (b) 1, 2, 22, 23, 24, ... (c) 1, -a, a2, -a3, a4, . . ., a ∈ R (d) 1, a2, a4, a6, ... , a ∈ R (e) a, a3, a5, a7, ... , a ∈ R (f) 0, 1, 2(2), 3(22), 4(23), . . .
How many 20-digit quaternary (0, 1, 2, 3) sequences are there where: (a) There is at least one 2 and an odd number of 0's? (b) No symbol occurs exactly twice? (c) No symbol occurs exactly three times? (d) There are exactly two 3's or none at all?
Determine the sequence generated by each of the following exponential generating functions.(a) f(x) = 3e3x(b) f(x) = 6e5x - 3e2x(c) f(x) = ex + x2(d) f(x) = e2x - 3x3 + 5x2 + 7x(e) f(x) = 1/(1-x)(f) f(x) = 3/(1 - 2x) + e
In each of the following, the function f(x) is the exponential generating function for the sequence a0, a1, a2, . . . , whereas the function g(x) is the exponential generating function for the sequence b0, b1, b2, . . . . Express g(x) in terms of fix) if (a) b3 = 3 bn = an, n ∈ N, n ≠ 3 (b) an
(a) For the ship in Example 9.28, how many signals use at least one flag of each color? (Solve this with an exponential generating function.)(b) Restate part (a) in an alternative way that uses the concept of an onto function.(c) How many signals are there in Example 9.28, where the total number of
Find the exponential generating function for the sequence 0!, 1!, 2!, 3!, ....
(a) Find the exponential generating function for the number of ways to arrange n letters, n > 0, selected from each of the following words. i) HAWAIIii) MISSISSIPPIiii) ISOMORPHISM(b) For section (ii) of part (a), what is the exponential generating function if the arrangement must contain at
Given the sequences a0, a1, a2, . . . and b0, b1,b2, ..., with exponential generating functions f(x), g(x), respectively, show that if h{x) = f(x)g(x), then h{x) is the exponential generating function of the sequence c0, c1, c2, . . ., where cn = ∑ni=o (n1)a1,bn-1 for each n > 0.
If a 20-digit ternary (0, 1, 2) sequence is randomly generated, what is the probability that: (a) It has an even number of l's? (b) It has an even number of l's and an even number of 2's? (c) It has an odd number of 0's? (d) The total number of 0's and l's is odd? (e) The total number of 0's and 1
Find the generating function for the sequences (a) 1, 2, 3, 3, 3, . . . ; (b) 1, 2, 3, 4, 4, 4, ; (c) 1, 4, 7, 10, 13, ....
(a) Find the generating function for the sequences (i) 0, 1,0, 0, 0, ... ; (ii) 0, 1, 1,1,1,... ; (iii) 0, 1, 2, 3, 4, . . .; (iv) 0, 1, 3, 6, 10, .... (b) Use result (iv) from part (a) to find a formula for ∑nk=1 k.
Continue the development of the ideas set forth in Example 9.31 and derive the formula ∑ni - 0= 0 i3 = [n(n + 1)/2]2
If f(x) = ∑∞n=0,anxn what is the generating function for the sequencer, a0 + a0, a1 + a2, a2 + a3, . . . ? What is the generating function for the sequence a0, a0 + a1, a2 + a2 + a1, a2 + a3 + a3, a2 + a3 + a4, . .. ? What is the generating function for the sequence a0/4 + a0/2, a1/4 + a1/2 +
Let f(x) be the generating function for the sequence a0, a1, a2, .... For what sequence is (1 - x)f(x) the generating function?
Let f(x) = ∑∞t = 0 a1x1 with f(1) = ∑∞t=0, a1 a finite number. Verfiy that the quotient [f(x) - f(1)]/(x-1) is the generating function for the sequence s0,s1,s2......., where sn = ∑∞t= n+1 at , n ∈ N.
Find the generating function for the sequence a0, a1, a2, ..., where an" = ∑ni=0 (1/i!) n ∈ N.
(a) Find the generating function for the sequence 0, 1, 3, 6, 10, 15, . . . (where 1, 3, 6, 10, 15, . . . are the triangular numbers of Example 4.5).(b) For n ∈ Z+, determine a formula for the sum of the first n triangular numbers.
Find the generating function for each of the following sequences. (a) 7, 8, 9, 10,. . . (b) 1, a, a2, a3, a4, ... , a ∈ R (c) 1, (1 + a), (1 + a)2, (1 + a)3, . . . , a ∈ R (d) 2, 1 + a , 1+ a2, 1 + a3, ........ a ∈ R
Determine the generating function for the number of partitions of n ∈ N where 1 occurs at most once, 2 occurs at most twice, 3 at most thrice, and, in general, k occurs at most k times, for every k ∈ Z+.
In a rural area 12 mailboxes are located at a general store.a) If a news carrier has 20 identical fliers, in how many ways can she distribute the fliers so that each mailbox gets at least one flier?b) If the mailboxes are in two rows of six each, what is the probability that a distribution from
Let S be a set containing n distinct objects. Verify that ex/(I - x)k is the exponential generating function for the number of ways to choose m of the objects in S, for 0 < m < ft, and distribute these objects among k distinct containers, with the order of the objects in any container relevant for
(a) For a, d ∈ R, find the generating function for the sequence a, a + d, a + 2d, a + 3d, ....(b) For n ∈ Z+, use the result from part (a) to find a formula for the sum of the first n terms of the arithmetic progression a, a -f- d, a -f- 2d, a -f- 3d, ....
(a) For the alphabet ∑ = {0, 1}, let an count the number of strings of length n in ∑- that is, for n ∈ N, an = |∑n|. Determine the generating function for the sequence a0, a1, a2, . . . .(b) Answer the question posed in part (a) when | ∑ | - k, a fixed positive integer.
Suppose that X is a discrete random variable with probability distribution given bywhere k is a constant. Determine (a) the value of k;(b) Pr (X = 3), Pr (X 3), Pr (X > 2); and(c) Pr (X > 4X > 2), Pr (X > 104X > 102).
Suppose that Y is a geometric random variable where the probability of success for each Bernoulli trial is p. If m, n ∈ Z+ with m > n, determine Pr (Y > m\Y > n).
A test car is driven a fixed distance of n miles along a straight highway. (Here n ∈ Z+.) The car travels at one mile per hour for the first mile, two miles per hour for the second mile, four miles per hour for the third mile, . . . , and 2n~l miles per hour for the nth mile.(a) What is the car's
Find the coefficient of x83 in f(x) = (x5 + x8 + x11 + x14 + x17)10.
Sergeant Bueti must distribute 40 bullets (20 for rifles and 20 for handguns) among four police officers so that each officer gets at least two, but no more than seven, bullets of each type. In how many ways can he do this?
For n ∈ Z+, show that the number of partitions of n in which no even summand is repeated (an odd summand may or may not be repeated) is the same as the number of partitions of n where no summand occurs more than three times.
How many 10-digit telephone numbers use only the digits 1, 3, 5 and 7, with each digit appearing at least twice or not at all?
(a) For what sequence of numbers is g(x) = (1 - 2x) -5/2 the exponential generating function?(b) Find a and b so that (1 - ax)b is the exponential generating function for the sequence 1,7, 7 • 11, 7 • 11 • 15, ....
For integers n, k > 0 let • P1 be the number of partitions of n. • P2 be the number of partitions of In + k, where n + k is the greatest summand. • P3 be the number of partitions of 2n + k into precisely n + k summands. Using the concept of the Ferrers graph, prove that P1 = P2 and P2 = P3,
Simplify the following sum where n ˆˆ Z+: (Hint: You may wish to start with the binomial theorem.)
In each of the following, f: Z+ → R. Solve for fin) relative to the given set S, and determine the appropriate "big-Oh" form for / on S. (a) f(1) = 5 f(n) = 4f(n/3) + 5, n = 3,9, 27, . .. S = {3ʹ|I ∈ N] (b) f(l) - 7 f(n) = f(n/5) + 7, n = 5, 25, 125, . . . 5 = {5ʹ|i ∈ N}
In this exercise we briefly introduce the Master Theorem. (For more on this result, including a proof, we refer the reader to pp. 73-84 of reference [5] by T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein.)Consider the recurrence relationf(D = l,f(n) = af(n/b) + h(n),where n ∈ Z+, n >
Let a, b, c e Z+ with b > 2, and let d ∈ N. Prove that the solution for the recurrence relation f(1) = d f(n) = af(n/b) + c, n = bk, k ≥ 1 satisfies (a) f(n) = d + c logb n, for n = bk, k ∈ N, when a = 1. (b) f(n) - dnlogb a + (c/(a - 1 ))[nlogb a - 1], for n = bk, k ∈ N, when a ≥ 2.
In each of the following, f: Z+ → R. Solve for f(n) relative to the given set S, and determine the appropriate "big-Oh" form for f on S. (a) f(l) = 0 F(n) = 2f(n/5) + 3, n = 5, 25, 125, . . . S = {5ʹ| i ∈ N} (b) f(1) = 1 f(n) = f(n/2) + 2, n = 2, 4, 8, . . . 5 = {2ʹ|i ∈ N}
Consider a tennis tournament for n players, where n = 2k, k ∈ Z+. In the first round n/2 matches are played, and the n/2 winners advance to round 2, where n/4 matches are played. This halving process continues until a winner is determined. (a) For n = 2k, k ∈ Z+, let f(n) count the total number
Complete the proofs for Corollary 10.1 and parts (b) and (c) of Theorem 10.2. Corollary 10.1 Let a,b, c ∈ Z+ with b ≥ 2, and let f: Z+ → R. If f(1) = c, and f(n)=af(n/b) + c, for n=bk, k ≥ 1, then (1) f ∈ 0(logb n) on {bk|k ∈ N}, when a = 1, and (2) f ∈ 0(nlogb a) on {bk|k ∈ N},
(a) Modify the procedure in Example 10.48 as follows: For any S ⊂ R, where |S| = n, partition S as S1 ∪ S2, where |S1| = |S2|, for n even, and |S1| = 1 + |S2|, for w odd. Show that if f(n) counts the number of comparisons needed (in this procedure) to find the maximum and minimum elements of S,
In Corollary 10.2 we were concerned with finding the appropriate "big-Oh" form for a function f: Z+ †’ R+ U {0} wheref(1) ‰¤ c, for c ˆˆ Z+f(n) ‰¤ af (n / b) + c,for a, b ˆˆ Z+ with b ‰¥ 2, and n = bk, k ˆˆ Z+.Here the
For n ˆˆ Z+ and n ‰¥ k + 1 ‰¥ 1, verify algebraically the recursion formula
Renu wants to sell her laptop for $4000. Narmada offers to buy it for $3000. Renu then splits the difference and asks for $3500. Narmada likewise splits the difference and makes a new offer of $3250. (a) If the women continue this process (of asking prices and counteroffers), what will Narmada be
Parts (a) and (b) of Fig. 10.27 provide the Hasse diagrams for two partial orders referred to as thq fences F5, F6 [on 5, 6 (distinct) elements, respectively]. If, for instance, R denotes the partial order for the fence F5, then a1 R a2, a3 R a2, a3 R a4, and a5 R a4. For each such fence Fn, n
For n ‰¥ 0, let m = [(n + 1)/2]. Prove that Fn+2 =(You may want to look back at Examples 9.17 and 10.11.)
(a) For n ∈ Z+, determine the number of ways one can tile a 1 × n chessboard using 1 × 1 white (square) tiles and 1×2 blue (rectangular) tiles.(b) How many of the tilings in part (a) use (i) no blue tiles; (ii) exactly one blue tile; (iii) exactly two blue tile&? (iv) exactly three blue
LetHow is c2 related to c? What is the value of c?
For n ˆˆ Z+, dn denotes the number of derangements of {1, 2, 3, . . ., n], as discussed in Section 8.3.(a) If n > 2, show that dn satisfies the recurrence relationdn = (n - l)(dn-1 + dn-2), d2 = l, d1 = 0.(b) How can we define d0 so that the result in part (a) is valid for n ‰¥
For n ≥ 0, draw n ovals in the plane so that each oval intersects each of the others in exactly two points and no three ovals are coincident. If an denotes the number of regions in the plane that results from these n ovals, find and solve a recurrence relation for an.
For n ≥ 0, let us toss a coin 2n times.(a) If an is the number of sequences of 2n tosses where n heads and n tails occur, find an in terms of n.(b) Find constants r, s, and t so that (r + sx)t = f(x) = ∑∞n=0 anxn.(c) Let bn denote the number of sequences of 2n tosses where the numbers of
For a = (1 + √5)/2 and β = (1 - √5)/2, show that ∑∞k=0 βk = - β and that ∑∞k=0 |β|= a2.
Let a, b, c be fixed real numbers with ab = 1 and let f:R × R → R be the binary operation, where f(x,y) = a + bxy + c(x + y). Determine the value(s) of c for which f will be associative.
(a) For n ‰¥ 0, let Bn denote the number of partitions of {1, 2, 3, . . . , n}. Set B0 = 1 for the partitions of 0. Verify that for all n ‰¥ 0,[The numbers B1,i ‰¥ 0, are referred to as the Bell numbers after Eric Temple Bell (1883-1960).](b) How are the Bell numbers related to the
(a) For a = (1 + √5)/2 and β = (1 - √5)/2, verify that a2 - a-2 = a - β = β-2 -β2. (b) Prove that F2n = F2n+l - F2n-1, n ≥ 1. (c) For n ≥ 1, let T be an isosceles trapezoid with bases of length Fn-1 and Fn+1, and sides of length Fn. Prove that the area of T is (√3/4)F2n.
Let S be the sample space for an experiment E. If A, B are events from S with A ∪ B = S, A ∪ B = ∅, Pr(A) = p, and Pr(B) = p2, determine p.
De'Jzaun and Sandra toss a loaded coin, where Pr(H) = p > 0. The first to obtain a head is the winner. Sandra goes first but, if she tosses a tail, then De'Jzaun gets two chances. If he tosses two tails, then Sandra again tosses the coin and, if her toss is a tail, then De'Jzaun again goes twice
For n ≥ 1, let an count the number of binary strings of length n, where there is no run of l's of odd length. Consequently, when n = 6, for instance, we want to include the strings 110000 (which has a run of two l's and a run of four 0's) and 011110 (which has two runs of one 0 and one run of
Let a, b be fixed nonzero real numbers. Determine xn if xn = xn- 1xn-2, n ≥ 2,x0 = a, x1 = b.
(a) Evaluate F2n+1 - FnFn+l - F2n for n = 0, 1, 2, 3. (b) From the results in part (a), conjecture a formula for F2n+l - FnFn+l-F2n for n ∈ N. (c) Establish the conjecture in part (b) using the Principle of Mathematical Induction.
Let n ∈ Z+. On a 1 × n chessboard two kings are called nontaking, if they do not occupy adjacent squares. In how many ways can one place 0 or more nontaking kings on a 1 × n chessboard?
(a) For 1 ≤ i ≤ 6, determine the rook polynomial r(C1, x) for the chessboard Ct shown in Fig. 10.28. (b) For each rook polynomial in part (a), find the sum of the coefficients of the powers of x -that is, determine r (Ct, 1) for 1 ≤ i ≤ 6.
(Gambler's Ruin) When Cathy and Jill play checkers, each has probability \ of winning. There is never a tie, and the games are independent in the sense that no matter how many games the girls have played, each girl still has probability 1/2 of winning the next game. After each game the loser gives
For n, m ∈ Z+, let f(n, m) count the number of partitions of n where the summands form a non increasing sequence of positive integers and no summand exceeds m. With n = 4 and m = 2, for example, we find that f(4, 2) = 3 because here we are concerned with the three partitions 4 = 2 + 2, 4 = 2+1
Let n,k ∈ Z+, and define p(n, k) to be the number of partitions of n into exactly k (positive-integer) summands. Prove that p(n, k) = p(n - 1), (k - 1) + p(n - k, k).
Let A, B be sets with |A| = m ‰¥ n = |B|, and let a(m, n) count the number of onto functions from A to B. Show thata(m, 1) = 1when m ‰¥ n > 1.
When one examines the units digit of each Fibonacci number Fn, n ≥ 0, one finds that these digits form a sequence that repeats after 60 terms. [This was first proved by Joseph-Louis Lagrange (1736-1813).] Write a computer program (or develop an algorithm) to calculate this sequence of 60 digits.
For n ≥ 1, let an count the number of ways to write n as an ordered sum of odd positive integers. (For example, a4 = 3 since 4 = 3 + l = l + 3 = l + l + l + l.) Find and solve a recurrence relation for an.
(a) Compute A2, A3, and A4.(b) Conjecture a general formula for An, n ˆˆ Z+, and establish your conjecture by the Principle of Mathematical Induction.Let
(a) Compute M2, M3, and M4.(b) Conjecture a general formula for Mn, n ˆˆ Z+, and establish your conjecture by the Principle of Mathematical Induction.Let
Determine the points of intersection of the parabola y = x2 - 1 and the hyperbola y = 1 + 1/x. .
Let a = (1 + ˆš5)/2 and Î² = (1 - ˆš5)/2.(a) Verify that a2 = a + 1 and Î²2 = Î² + 1.(c) Show that a3 = 1 + 2a and Î²3 = 1 + 2Î².
(a) For a = (1 + ˆš5)/2, verify that a2 + 1 = 2 + a and (2 + a)2 = 5a2.(b) Show that for Î² = (1 - ˆš5)/2, Î²2 + 1 = 2 + Î² and (2 + Î²)2 = 5Î²2.(c) If n, m ˆˆ N prove that
List three situations, different from those in this section, where a graph could prove useful
(a) If G = (V, E) is an undirected graph with |V| = v, |E| = e, and no loops, prove that 2e ≤ v2 - v.(b) State the corresponding inequality for the case when G is directed.
(a) Consider the three connected undirected graphs in Fig. 11.11. The graph in part (a) of the figure consists of a cycle (on the vertices u1, u2, u3) and a vertex u4 with edges (spokes) drawn from u4 to the other three vertices. This graph is called the wheel with three spokes and is denoted by
For the undirected graph in Fig. 11.12, find and solve a recurrence relation for the number of closed v.v walks of length n ‰¥ 1, if we allow such a walk, in this case, to contain or consist of one or more loops.
Unit-Interval Graphs. For n ‰¥ 1, we start with n closed intervals of unit length and draw the corresponding unit-interval graph on n vertices, as shown in Fig. 11.13. In part (a) of the figure we have one unit interval. This corresponds to the single vertex u; both the interval and the
For the graph in Fig. 11.7, determine(a) A walk from b to d that is not a trail;(b) A b-d trail that is not a path;(c) A path from b to d;(d) A closed walk from b to b that is not a circuit;(e) A circuit from b to b that is not a cycle; and(f) A cycle from b to b.
For n ≥ 2, let G = (V, E) be the loop-free undirected graph, where V is the set of binary n-tuples (of 0's and l's) and E = {{v, w}|v, w ∈ V and v, w differ in (exactly) two positions}. Find k(G).
Let G = (V, E) be the undirected graph in Fig. 11.8. How many paths are there in G from a to h? How many of these paths have length 5?
If a, b are distinct vertices in a connected undirected graph G, the distance from a to b is defined to be the length of a shortest path from a to b (when a = b the distance is defined to be 0). For the graph in Fig. 11.9, find the distances from d to (each of) the other vertices in G.
Seven towns a, b,c, d, e, f, and g are connected by a system of highways as follows: (1) 1-22 goes from a to c, passing through b; (2) 1-33 goes from c to d and then passes through b as it continues to f; (3) 1-44 goes from d through e to a; (4) 1-55 goes from f to b, passing through g; and (5)
Let G = (V, E) be a loop-free connected undirected graph, and let {a, b} be an edge of G. Prove that {a, b} is part of a cycle if and only if its removal (the vertices a and b are left) does not disconnect G.
Let G be the undirected graph in Fig. 11.27(a).(a) How many connected subgraphs of G have four vertices and include a cycle?(b) Describe the subgraph G1 (of G) in part (b) of the figure first, as an induced subgraph and second, in terms of deleting a vertex of G.(c) Describe the subgraph G2 (of G)
(a) If G1, G2 are (loop-free) undirected graphs, prove that G1, G2 are isomorphic if and only if 1, 2 are isomorphic.(b) Determine whether the graphs in Fig. 11.30 are isomorphic.
(a) Let G be an undirected graph with n vertices. If G is isomorphic to its own complement , how many edges must G have? (Such a graph is called self-complementary.) (b) Find an example of a self-complementary graph on four vertices and one on five vertices. (c) If G is a self-complementary graph
Let G be a cycle on n vertices. Prove that G is self- complementary if and only if n = 5.
(a) Find a graph G where both G and are connected. (b) If G is a graph on n vertices, for n ≥ 2, and G is not connected, prove that is connected.
(a) Extend Definition 11.13 to directed graphs.(b) Determine whether the directed graphs in Fig. 11.31 are isomorphic
(a) How many subgraphs H = (V, E) of K6 satisfy |V| =3? (If two subgraphs are isomorphic but have different vertex sets, consider them distinct.) (b) How many subgraphs H = (V, E) of K6 satisfy |V| =4? (c) How many subgraphs does K6 have? (d) For n ≥ 3, how many subgraphs does Kn have?
Let v, w be two vertices in Kn, n ≥ 3. How many walks of length 3 are there from v to w?
(a) Let G = (V, E) be an undirected graph, with G1 = (V1, E1) a subgraph of G. Under what condition(s) is G1 not an induced subgraph of G? (b) For the graph G in Fig. 11.27(a), find a subgraph that is not an induced subgraph.

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