Introductory Combinatorics
Introductory Combinatorics
5th Edition
ISBN: 9780136020400
Author: Richard A. Brualdi
Publisher: Prentice Hall
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Chapter 5, Problem 19E
To determine

The sum the series 12+22+32+...+n2 by observing that m2=2(m2)+(m1) and by using the identity (5.19).

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Chapter 5 Solutions

Introductory Combinatorics

Ch. 5 - Use combinatorial reasoning to prove the identity...Ch. 5 - Let n be a positive integer. Prove that (Hint:...Ch. 5 - Find one binomial coefficient equal to the...Ch. 5 - Prob. 14ECh. 5 - Prove, that for every integer n > 1, Ch. 5 - By integrating the binomial expansion, prove that,...Ch. 5 - Prob. 17ECh. 5 - Evaluate the sum Ch. 5 - Sum the series by observing that and using the...Ch. 5 - Find integers a, b, and c such that for all m....Ch. 5 - Prob. 21ECh. 5 - Prob. 22ECh. 5 - Prob. 23ECh. 5 - Prob. 24ECh. 5 - Use a combinatorial argument to prove the...Ch. 5 - Let n and k be integers with 1 ≤ k ≤ n. Prove...Ch. 5 - Let n and k be positive integers. Give a...Ch. 5 - Let n and k be positive integers. Give a...Ch. 5 - Find and prove a formula for where the summation...Ch. 5 - Prove that the only antichain of S = {1, 2, 3, 4}...Ch. 5 - Prove that there are only two antichains of S =...Ch. 5 - Let S be a set of n elements. Prove that, if n is...Ch. 5 - Construct a partition of the subsets of {1, 2, 3,...Ch. 5 - In a partition of the subsets of {1,2, …, n} into...Ch. 5 - A talk show host has just bought 10 new jokes....Ch. 5 - Prove the identity of Exercise 25 using the...Ch. 5 - Use the multinomial theorem to show that, for...Ch. 5 - Use the multinomial theorem to expand (x1 + x2 +...Ch. 5 - Determine the coefficient of in the expansion...Ch. 5 - What is the coefficient of in the expansion of Ch. 5 - Prob. 41ECh. 5 - Prob. 42ECh. 5 - Prove by induction on n that, for n a positive...Ch. 5 - Prove that where the summation extends over all...Ch. 5 - Prove that where the summation extends over all...Ch. 5 - Use Newton’s binomial theorem to approximate . Ch. 5 - Use Newton’s binomial theorem to approximate...Ch. 5 - Use Theorem 5.6.1 to show that, if m and n are...Ch. 5 - Use Theorem 5.6.1 to show that, if m and n are...Ch. 5 - Prob. 50ECh. 5 - Let R and S be two partial orders on the same set...
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