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Questions and Answers of
Linear Algebra
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)
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
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, ... ,
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
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)
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
(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
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
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
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
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
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 +
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=
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
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
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
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
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
(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
(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)
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
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
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
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
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,
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
Simplify the following sum where n 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 =
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.
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,
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 =
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
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 ≥
(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
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 Z+f(n) ¤ af (n /
For n 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.
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
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
LetHow is c2 related to c? What is the value of c?
For n 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),
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
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 =
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
(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
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
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
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
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
(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
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
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
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
(a) Compute A2, A3, and A4.(b) Conjecture a general formula for An, n 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 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 + 5)/2 and β = (1 - 5)/2.(a) Verify that a2 = a + 1 and β2 = β + 1.(c) Show that a3 = 1 + 2a and β3 = 1 +
(a) For a = (1 + 5)/2, verify that a2 + 1 = 2 + a and (2 + a)2 = 5a2.(b) Show that for β = (1 - 5)/2, β2 + 1 = 2 + β and (2 +
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
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
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
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
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
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
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
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
(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
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|
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
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