Multiple Choice Questions
Advanced Discrete Mathematics: Graph Theory and Combinatorics
Topic: Graph Theory
Grade: 11
Question 1:
Which of the following is NOT a valid graph?
Answer Choices:
A) Complete graph
B) Bipartite graph
C) Eulerian graph
D) Hamiltonian graph
Answer: C) Eulerian graph
Explanation: An Eulerian graph is a graph that contains an Eulerian circuit, which is a closed walk that visits each edge exactly once. However, an Eulerian circuit is only possible if the graph is connected and all vertices have even degree. Therefore, if a graph is not connected or has vertices with odd degree, it cannot be Eulerian. For example, consider a graph with two disconnected components, each containing an odd number of vertices. Such a graph would not be Eulerian.
Question 2:
In a simple undirected graph with 10 vertices, what is the maximum number of edges possible?
Answer Choices:
A) 10
B) 20
C) 45
D) 90
Answer: C) 45
Explanation: The maximum number of edges in a simple undirected graph with n vertices is given by the formula (n * (n-1))/2. Plugging in n = 10, we get (10 * 9)/2 = 45. This can be visualized by considering a complete graph with 10 vertices, where each vertex is connected to every other vertex. For example, a complete graph with 4 vertices would have (4 * 3)/2 = 6 edges.
Topic: Combinatorics
Question 3:
How many ways can 5 people be seated in a row of 10 chairs if 2 of the people refuse to sit next to each other?
Answer Choices:
A) 120
B) 240
C) 360
D) 720
Answer: B) 240
Explanation: To solve this problem, we can consider the two people who refuse to sit together as a single entity. We can arrange the remaining 4 people and this entity in 5! = 120 ways. Within this arrangement, the two people who refuse to sit together can be arranged in 2! = 2 ways. Therefore, the total number of arrangements is 120 * 2 = 240. For example, consider the arrangement ABXCD, where A, B, C, and D are the remaining 4 people and X represents the entity of the two people who refuse to sit together. This arrangement satisfies the given condition.
Question 4:
How many distinct permutations can be formed using all the letters of the word \”MISSISSIPPI\”?
Answer Choices:
A) 34650
B) 2520
C) 840
D) 420
Answer: A) 34650
Explanation: The word \”MISSISSIPPI\” has 11 letters, with 1 M, 4 Is, 4 Ss, and 2 Ps. The number of distinct permutations can be calculated using the formula n! / (n1! * n2! * … * nk!), where n is the total number of letters and n1, n2, …, nk are the frequencies of each letter. Plugging in the values, we get 11! / (1! * 4! * 4! * 2!) = 34650. For example, consider the permutation \”ISSMPSISIPSI\”, which satisfies the given conditions.
Topic: Graph Theory
Question 5:
In a simple connected graph with n vertices, the maximum number of edges is given by:
Answer Choices:
A) n
B) n-1
C) n(n-1)/2
D) 2n
Answer: C) n(n-1)/2
Explanation: In a simple connected graph with n vertices, each vertex can be connected to every other vertex except itself. Therefore, the maximum number of edges is given by the formula n(n-1)/2. For example, in a graph with 4 vertices, the maximum number of edges is 4(4-1)/2 = 6.
Question 6:
A connected graph with n vertices and n-1 edges is called:
Answer Choices:
A) Complete graph
B) Bipartite graph
C) Eulerian graph
D) Tree
Answer: D) Tree
Explanation: A connected graph with n vertices and n-1 edges is called a tree. A tree is a special type of graph where there is exactly one path between any two vertices. It is also a connected acyclic graph. For example, consider a graph with 5 vertices and 4 edges, where each edge connects two vertices. This graph is a tree.
Topic: Combinatorics
Question 7:
How many 3-digit numbers can be formed using the digits 1, 2, 3, and 4 without repetition?
Answer Choices:
A) 12
B) 24
C) 36
D) 48
Answer: C) 36
Explanation: Since repetition is not allowed, the first digit can be chosen from 4 options, the second digit from 3 options, and the third digit from 2 options. Therefore, the total number of 3-digit numbers is 4 * 3 * 2 = 24. For example, consider the numbers 123, 132, 213, 231, 312, and 321, which are all valid 3-digit numbers that can be formed using the given digits.
Question 8:
How many ways can 6 people be seated in a row of 6 chairs if 2 of the people must sit together?
Answer Choices:
A) 240
B) 360
C) 480
D) 720
Answer: A) 240
Explanation: To solve this problem, we can consider the two people who must sit together as a single entity. We can arrange the remaining 4 people and this entity in 5! = 120 ways. Within this arrangement, the two people who must sit together can be arranged in 2! = 2 ways. Therefore, the total number of arrangements is 120 * 2 = 240. For example, consider the arrangement ABCDEY, where A, B, C, D, E are the remaining 4 people and Y represents the entity of the two people who must sit together. This arrangement satisfies the given condition.