Introductory Combinatorics
5th Edition
ISBN: 9780136020400
Author: Richard A. Brualdi
Publisher: Prentice Hall
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Chapter 7, Problem 38E
(a)
To determine
To solve: The recurrence relation by investigative the first few values for a formula and then proving the conjectured formula by induction of
(b)
To determine
To solve: The recurrence relation by investigative the first few values for a formula and then proving the conjectured formula by induction of
(c)
To determine
To solve: The recurrence relation by investigative the first few values for a formula and then proving the conjectured formula by induction of
(d)
To determine
To solve: The recurrence relation by investigative the first few values for a formula and then proving the conjectured formula by induction of
(e)
To determine
To solve: The recurrence relation by investigative the first few values for a formula and then proving the conjectured formula by induction of
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Chapter 7 Solutions
Introductory Combinatorics
Ch. 7 - Prob. 1ECh. 7 - Prove that the nth Fibonacci number fn is the...Ch. 7 - Prove the following about the Fibonacci...Ch. 7 - 4. Prove that the Fibonacci sequence is the...Ch. 7 - By examining the Fibonacci sequence, make a...Ch. 7 - * Let m and n be positive integers. Prove that if...Ch. 7 - * Let m and n be positive integers whose greatest...Ch. 7 - Consider a 1-by-n chessboard. Suppose we color...Ch. 7 - Prob. 9ECh. 7 - Prob. 10E
Ch. 7 - Prob. 11ECh. 7 - Prob. 12ECh. 7 - 13. Determine the generating function for each of...Ch. 7 - 14. Let S be the multiset {∞ · e1, ∞ · e2, ∞ · e3,...Ch. 7 - 15. Determine the generating function for the...Ch. 7 - 16. Formulate a combinatorial problem for which...Ch. 7 - 17. Determine the generating function for the...Ch. 7 - 18. Determine the generating function for the...Ch. 7 - 19. Let h0, h1, h2, …, hn, … be the sequence...Ch. 7 - Prob. 20ECh. 7 - 21. * Let hn denote the number of regions into...Ch. 7 - 22. Determine the exponential generating function...Ch. 7 - 23. Let α be a real number. Let the sequence h0,...Ch. 7 - 24. Let S be the multiset {∞ · e1, ∞ · e2, · , ∞ ·...Ch. 7 - 25. Let hn denote the number of ways to color the...Ch. 7 - Determine the number of ways to color the squares...Ch. 7 - Determine the number of n-digit numbers with all...Ch. 7 - Determine the number of n-digit numbers with all...Ch. 7 - We have used exponential generating functions to...Ch. 7 - Prob. 30ECh. 7 - Solve the recurrence relation hn = 4hn−2, (n ≥ 2)...Ch. 7 - Prob. 32ECh. 7 - Solve the recurrence relation hn = hn−1 + 9hn−2 −...Ch. 7 - Solve the recurrence relation hn = 8hn−1 − 16hn−2,...Ch. 7 - Solve the recurrence relation hn = 3hn − 2 − 2hn −...Ch. 7 - Prob. 36ECh. 7 - Determine a recurrence relation for the number an...Ch. 7 - Prob. 38ECh. 7 - Let hn denote the number of ways to perfectly...Ch. 7 - Let an equal the number of ternary strings of...Ch. 7 - * Let 2n equally spaced points be chosen on a...Ch. 7 - Solve the nonhomogeneous recurrence relation
Ch. 7 - Solve the nonhomogeneous recurrence relation
hn =...Ch. 7 - Solve the nonhomogeneous recurrence relation
Ch. 7 - Prob. 45ECh. 7 - Solve the nonhomogeneous recurrence relation
Ch. 7 - Solve the nonhomogeneous recurrence relation
Ch. 7 - Solve the following recurrence relations by using...Ch. 7 - (q-binomial theorem) Prove that
where
is the...Ch. 7 - Call a subset S of the integers {1, 2, …, n}...Ch. 7 - Solve the recurrence relation
from Section 7.6...Ch. 7 - Prob. 52ECh. 7 - Suppose you deposit $500 in a bank account that...
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- 1. (20 pts) Determine whether the following statements are true (T) or false (F)? (A reasoning is required.) (1) Let V be the set of all ordered pairs of real numbers. Consider the following addition and scalar multiplication operations on u = u= (u1, u2) and v = (v1, v2): u + v = (U₁ + V₁, U₂ + v₂), ku = (ku₁, u₂). Is V a vector space under the above operations? U2 (2) The set Mmxn of all m×n matrices with the usual operations of addition and scalar multiplication is a vector space. α (3) The dimension of the vector space of all matrices A = [a b] in R2×2 with a+d=0 is 4. (4) The coordinate vector of p(x) = 2-x+x² in P3 relative to the basis S = {1, 1+x, x + x2} is [4 -2 1]. (5) If a 6×4 matrix A has a rank 3, then the dimension of N(A) is 3.arrow_forwardScenario Sales of products by color follow a peculiar, but predictable, pattern that determines how many units will sell in any given year. This pattern is shown below Product Color 1995 1996 1997 Red 28 42 21 1998 23 1999 29 2000 2001 2002 Unit Sales 2003 2004 15 8 4 2 1 2005 2006 discontinued Green 26 39 20 22 28 14 7 4 2 White 43 65 33 36 45 23 12 Brown 58 87 44 48 60 Yellow 37 56 28 31 Black 28 42 21 Orange 19 29 Purple Total 28 42 21 49 68 78 95 123 176 181 164 127 24 179 Questions A) Which color will sell the most units in 2007? B) Which color will sell the most units combined in the 2007 to 2009 period? Please show all your analysis, leave formulas in cells, and specify any assumptions you make.arrow_forward5. (20%) The linear transformation L: P3 → P2 defined by L(f(x)) = f'(x)+ f(0). (a) Find the representing matrix A of L with respect to the ordered basis {x2, x, 1} for P3, and the ordered basis {2,1 - x} for P2. (b) Find the coordinates of the f(x) = 2x² +2 in P3 with respect to the ordered basis {x2,-x, 1}, and find the coordinates of L(f(x)) with respect to the ordered basis {2,1-x}arrow_forward
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