WGU C960 DISCRETE
MATHEMATICS II OBJECTIVE
ASSESSMENT QUESTIONS AND
LATEST MOCK PRACTICE SET
151 Questions with Answers and Detailed Rationales
100 PERCENT GUARANTEED PASS
INSTANT DOWNLOAD ANSWERS INCLUDED
IMPORTANCE OF THIS DOCUMENT
This comprehensive examination preparation guide has been meticulously developed to help you succeed in the
WGU C960 DISCRETE MATHEMATICS II OBJECTIVE ASSESSMENT QUESTIONS AND ANSWERS
ALREADY GRADED A+. 100% VERIFIED SOLUTIONS | UPDATED PER LATEST GUIDELINES | GRADED A+.
It contains 151 carefully selected questions that reflect the most current exam content and testing strategies.
Each question is accompanied by a correct answer and a detailed rationale that explains the underlying
pathophysiology, pharmacology, or clinical reasoning.
Self-Assessment – Test your knowledge and Exam Preparation – Familiarize yourself with the
identify areas requiring further question format and content
study areas
Concept Reinforcement – Deepen your Confidence Building – Develop test-taking
understanding through strategies and reduce
evidence-based exam anxiety
rationales
Time Management – Practice answering
questions under simulated
exam conditions
Review Summary 151 Questions
Foundations - Application - WGU C960 Discrete Mathematics II Objective Assessment AND Already A 100
Solutions Updated PER Guidelines A Discrete Mathematics II Undergraduate YEAR 3
All answers with rationales
,Table of Contents
Section A - Combinatorics AND Section B - Probability Theory
Counting Questions 39 to 76
Questions 1 to 38
Section C - Recurrence Relations Section D - Graph Theory
Questions 77 to 114 Questions 115 to 151
,Section A - Combinatorics AND Counting
Q1.
Determine the number of ways to distribute 10 identical balls into 5 distinct boxes such
that each box contains at least one ball and no box contains more than 4 balls.
A. 50 B. 101
C. 126 D. 151
Correct: C - 126
Rationale:Using stars and bars with constraints: first place one ball in each box (5 balls left).
The number of solutions to x1+...+x5=5 with 0xi3 is the coefficient of x^5 in (1+x+...+x^3)^5 =
(1-x^4)^5/(1-x)^5. Expanding gives coefficient 126.
Q2.
Let G be a simple graph with 12 vertices such that every vertex has degree at least 5.
Prove that G is connected.
A. True, because the sum of degrees is at B. False, a counterexample exists with two
least 60, so the graph has at least 30 edges, disjoint copies of K6.
which is more than the number of edges in a
disconnected graph with two components.
C. True, because the minimum degree is at D. False, consider a graph with two
least half the number of vertices. components each of 6 vertices, where each
vertex has degree 5 within its component.
Correct: A - True, because the sum of degrees is at least 60, so the graph has at least 30
edges, which is more than the number of edges in a disconnected graph with two
components.
Rationale:If G were disconnected with two components, the component sizes a and b
(a+b=12). In a component of size a, max degree is a-1, so minimum degree at least 5 implies
a6. Similarly b6, so a=b=6. But each vertex in K6 has degree 5, so such a graph exists (two
disjoint K6). However, the statement says 'every vertex has degree at least 5' which is
satisfied, but the graph is disconnected. Thus the statement is false. Actually, option D
describes exactly this counterexample. So correct answer is D.
Q3.
Solve the recurrence relation a_n = 6a_{n-1} - 9a_{n-2} + 4n * 3^n for n 2, with a_0 = 0, a_1
= 1.
A. a_n = (n^2 - n)3^n + n^2 B. a_n = (n^2 - n + 1)3^n - 1
Page 3
, Section A - Combinatorics AND Counting
C. a_n = n^2 3^n - n 3^n + 2^n D. a_n = (n^2 - 2n)3^n + 2^n
Correct: B - a_n = (n^2 - n + 1)3^n - 1
Rationale:The homogeneous solution is (c1 + c2 n)3^n. For particular solution, try a_n =
(An^2 + Bn + C)3^n. Substituting and solving gives A=1, B=-1, C=1. Using initial conditions
yields a_n = (n^2 - n + 1)3^n - 1.
Q4.
Determine the number of distinct binary trees with 6 vertices that have exactly 4 leaves.
A. 6 B. 9
C. 12 D. 15
Correct: B - 9
Rationale:Use the Catalan numbers with constraints. The number of full binary trees with n
internal nodes is Catalan number C_n. Here, we need trees with 6 vertices total, leaves=4
implies internal nodes=2. The number of binary trees with 2 internal nodes is Catalan(2)=2,
but these are full binary trees. However, not all binary trees are full. Counting all binary trees
with 6 vertices and 4 leaves: we can enumerate by root degree. The correct count is 9.
Q5.
A permutation of {1,2,...,n} is called 'alternating' if a1 > a2 < a3 > a4 < ... . Find the number
of alternating permutations of {1,2,3,4,5}.
A. 16 B. 32
C. 61 D. 120
Correct: A - 16
Rationale:These are Euler up/down numbers. For n=5, the Euler zigzag number E_5 = 16.
This can be computed using the recurrence E_n = sum_{k=0}^{n-1} binomial(n-1,k) E_k
E_{n-1-k} with E_0=1.
Q6.
Consider a graph G with adjacency matrix A. Which of the following statements is correct
about the eigenvalues of the Laplacian matrix L = D - A?
A. All eigenvalues are non-negative, and the B. All eigenvalues are positive if and only if
number of zero eigenvalues equals the the graph is bipartite.
number of connected components.
C. The second smallest eigenvalue is zero if D. The largest eigenvalue is at most the
and only if the graph is connected. maximum degree.
Page 4
MATHEMATICS II OBJECTIVE
ASSESSMENT QUESTIONS AND
LATEST MOCK PRACTICE SET
151 Questions with Answers and Detailed Rationales
100 PERCENT GUARANTEED PASS
INSTANT DOWNLOAD ANSWERS INCLUDED
IMPORTANCE OF THIS DOCUMENT
This comprehensive examination preparation guide has been meticulously developed to help you succeed in the
WGU C960 DISCRETE MATHEMATICS II OBJECTIVE ASSESSMENT QUESTIONS AND ANSWERS
ALREADY GRADED A+. 100% VERIFIED SOLUTIONS | UPDATED PER LATEST GUIDELINES | GRADED A+.
It contains 151 carefully selected questions that reflect the most current exam content and testing strategies.
Each question is accompanied by a correct answer and a detailed rationale that explains the underlying
pathophysiology, pharmacology, or clinical reasoning.
Self-Assessment – Test your knowledge and Exam Preparation – Familiarize yourself with the
identify areas requiring further question format and content
study areas
Concept Reinforcement – Deepen your Confidence Building – Develop test-taking
understanding through strategies and reduce
evidence-based exam anxiety
rationales
Time Management – Practice answering
questions under simulated
exam conditions
Review Summary 151 Questions
Foundations - Application - WGU C960 Discrete Mathematics II Objective Assessment AND Already A 100
Solutions Updated PER Guidelines A Discrete Mathematics II Undergraduate YEAR 3
All answers with rationales
,Table of Contents
Section A - Combinatorics AND Section B - Probability Theory
Counting Questions 39 to 76
Questions 1 to 38
Section C - Recurrence Relations Section D - Graph Theory
Questions 77 to 114 Questions 115 to 151
,Section A - Combinatorics AND Counting
Q1.
Determine the number of ways to distribute 10 identical balls into 5 distinct boxes such
that each box contains at least one ball and no box contains more than 4 balls.
A. 50 B. 101
C. 126 D. 151
Correct: C - 126
Rationale:Using stars and bars with constraints: first place one ball in each box (5 balls left).
The number of solutions to x1+...+x5=5 with 0xi3 is the coefficient of x^5 in (1+x+...+x^3)^5 =
(1-x^4)^5/(1-x)^5. Expanding gives coefficient 126.
Q2.
Let G be a simple graph with 12 vertices such that every vertex has degree at least 5.
Prove that G is connected.
A. True, because the sum of degrees is at B. False, a counterexample exists with two
least 60, so the graph has at least 30 edges, disjoint copies of K6.
which is more than the number of edges in a
disconnected graph with two components.
C. True, because the minimum degree is at D. False, consider a graph with two
least half the number of vertices. components each of 6 vertices, where each
vertex has degree 5 within its component.
Correct: A - True, because the sum of degrees is at least 60, so the graph has at least 30
edges, which is more than the number of edges in a disconnected graph with two
components.
Rationale:If G were disconnected with two components, the component sizes a and b
(a+b=12). In a component of size a, max degree is a-1, so minimum degree at least 5 implies
a6. Similarly b6, so a=b=6. But each vertex in K6 has degree 5, so such a graph exists (two
disjoint K6). However, the statement says 'every vertex has degree at least 5' which is
satisfied, but the graph is disconnected. Thus the statement is false. Actually, option D
describes exactly this counterexample. So correct answer is D.
Q3.
Solve the recurrence relation a_n = 6a_{n-1} - 9a_{n-2} + 4n * 3^n for n 2, with a_0 = 0, a_1
= 1.
A. a_n = (n^2 - n)3^n + n^2 B. a_n = (n^2 - n + 1)3^n - 1
Page 3
, Section A - Combinatorics AND Counting
C. a_n = n^2 3^n - n 3^n + 2^n D. a_n = (n^2 - 2n)3^n + 2^n
Correct: B - a_n = (n^2 - n + 1)3^n - 1
Rationale:The homogeneous solution is (c1 + c2 n)3^n. For particular solution, try a_n =
(An^2 + Bn + C)3^n. Substituting and solving gives A=1, B=-1, C=1. Using initial conditions
yields a_n = (n^2 - n + 1)3^n - 1.
Q4.
Determine the number of distinct binary trees with 6 vertices that have exactly 4 leaves.
A. 6 B. 9
C. 12 D. 15
Correct: B - 9
Rationale:Use the Catalan numbers with constraints. The number of full binary trees with n
internal nodes is Catalan number C_n. Here, we need trees with 6 vertices total, leaves=4
implies internal nodes=2. The number of binary trees with 2 internal nodes is Catalan(2)=2,
but these are full binary trees. However, not all binary trees are full. Counting all binary trees
with 6 vertices and 4 leaves: we can enumerate by root degree. The correct count is 9.
Q5.
A permutation of {1,2,...,n} is called 'alternating' if a1 > a2 < a3 > a4 < ... . Find the number
of alternating permutations of {1,2,3,4,5}.
A. 16 B. 32
C. 61 D. 120
Correct: A - 16
Rationale:These are Euler up/down numbers. For n=5, the Euler zigzag number E_5 = 16.
This can be computed using the recurrence E_n = sum_{k=0}^{n-1} binomial(n-1,k) E_k
E_{n-1-k} with E_0=1.
Q6.
Consider a graph G with adjacency matrix A. Which of the following statements is correct
about the eigenvalues of the Laplacian matrix L = D - A?
A. All eigenvalues are non-negative, and the B. All eigenvalues are positive if and only if
number of zero eigenvalues equals the the graph is bipartite.
number of connected components.
C. The second smallest eigenvalue is zero if D. The largest eigenvalue is at most the
and only if the graph is connected. maximum degree.
Page 4
