Solutions for Introductory Combinatorics
Problem 8E:
Use the pigeonhole principle to prove that the decimal expansion of a rational number m/n eventually...Problem 10E:
A child watches TV at least one hour each day for seven weeks but, because of parental rules, never...Problem 11E:
A student has 37 days to prepare for an examination. From past experience she knows that she will...Problem 12E:
Show by example that the conclusion of the Chinese remainder theorem (Application 6) need not hold...Problem 13E:
*Let S be a set of six points in the plane, with no three of the points collinear. Color either red...Problem 15E:
Prove that, for any n + 1 integers a1, a2,…,an+1, there exist two of the integers ai and aj with i ≠...Problem 17E:
There are 100 people at a party. Each person has an even number (possibly zero) of acquaintances....Problem 18E:
Prove that of any five points chosen within a square of side length 2, there are two whose distance...Problem 19E:
Prove that of any five points chosen within an equilateral triangle of side length 1, there are two...Problem 20E:
Prove that r(3, 3, 3) ≤ 17.
Problem 21E:
Prove that r(3, 3, 3) ≥ 17 by exhibiting a coloring, with colors red, blue, and green, of the line...Problem 27E:
A collection of subsets of {1, 2, …, n} has the property that each pair of subsets has at least one...Browse All Chapters of This Textbook
Chapter 1 - What Is Combinatorics?Chapter 2 - Permutations And CombinationsChapter 3 - The Pigeonhole PrincipleChapter 4 - Generating Permutations And CombinationsChapter 5 - The Binomial CoefficientsChapter 6 - The Inclusion-exclusion Principle And ApplicationsChapter 7 - Recurrence Relations And Generating FunctionsChapter 8 - Special Counting SequencesChapter 9 - Systems Of Distinct RepresentativesChapter 10 - Combinatorial Designs
Sample Solutions for this Textbook
We offer sample solutions for Introductory Combinatorics homework problems. See examples below:
To show this, we have to add 3 cases here. Case 1: Assume that one of m and n is even and the second...Procedure used: Multiplication principle: When a task has p outcomes and, no matter what the outcome...Given: The cumulative number of games played on the first n days is denoted by an, where n=1,2,…,77....Algorithm used: Begin with 1←,2←,⋯,n←. While there exists a mobile integer, do the following: (1)...Formula used: The pascal’s triangle formula is: (nk)=n!k!(n−k)!=n(n−1)⋅⋅⋅(n−k+1)k(k−1)⋅⋅⋅1...Suppose the set S={1,2,...,104}. Let A, B, C be the set of integers S that are divisible by 4, 5, 6...Using the mathematical induction and the Fibonacci recurrence. The sequence of numbers...Chapter 8, Problem 1EDefinition used: Let Y be a finite set and A=(A1,A2,…,An) be a family of n subsets of Y. A family...
Definition used: “Let n be a positive integer with n≥2, then Zn={0,1,…,n−1}.” “For any two integers...Definition used: “Two general graphs G=(V,E) and G=(V′,E′) are called isomorphic, provided that...Definition used: Chromatic number: Let G=(V,E) be a graph. A vertex coloring of G is an assignment...The given permutations are, f=(123456642153) and g=(123456356241). Here, (f∘g)(1)=2, (f∘g)(2)=5,...
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