Permutations And Combinations

Let the number of vertices
Given degree of each vertex
Then, total degree of simple graph
We know that,
sum of all degree of simple graph
number of edges in simple graph



The sum of the degrees of all vertices in a simple graph is equal to twice the number of edges. Therefore, in this case, the sum of the degrees of all vertices is 2*24 = 48.

Since the degree of each vertex is 3, the number of vertices can be found by dividing the sum of the degrees of all vertices by the degree of each vertex, i.e., 48/3 = 16.

Therefore, the number of vertices in the graph is 16.

The student needs to choose 10 questions, at least 4 from part A and 4 from part B. Let’s break down the selection process:

• There are 6 questions in part A and 7 questions in part B.
• The student must choose at least 4 questions from part A and at least 4 questions from part B. Therefore, the student must choose either 4, 5, or 6 questions from part A, and the remaining questions will be from part B.

We will now calculate the number of ways the student can choose questions:
1. Choose 4 questions from part A and 6 questions from part B:
2. Choose 5 questions from part A and 5 questions from part B:
3. Choose 6 questions from part A and 4 questions from part B: Now, add up all the possibilities:

The correct answer is (C) : 266.

The correct answer is: (D):

We are tasked with determining the number of possible ways in which three women and two men can occupy ten numbered chairs under the given conditions:

• The women choose chairs marked 1 to 6.
• The men choose from the remaining chairs.

Step 1: Women choosing their chairs

There are 6 chairs available for the women (chairs marked 1 to 6), and they need to choose 3 of these chairs. The number of ways in which the women can choose 3 chairs from the 6 available chairs is given by a permutation, since the order in which the women sit matters (as each woman will occupy a different chair). The number of ways is:

Step 2: Men choosing their chairs

After the women have chosen their chairs, 3 chairs are occupied, leaving 7 chairs (marked 7 to 10) for the men. The men must choose 2 of these remaining chairs. The number of ways in which the men can choose 2 chairs from the remaining 4 chairs is also a permutation, since the order in which the men sit matters. The number of ways is:

Step 3: Total number of possible ways

The total number of possible ways is the product of the two permutations (for the women and the men), so the total number of ways is:

Conclusion:
The correct answer is (D): .

We are asked to form four-digit even numbers using the digits 0, 1, 2, and 3 without repetition.
- The first digit cannot be 0 (since it would not be a four-digit number), so the first digit can be 1, 2, or 3. This gives us 3 choices for the first digit.
- The last digit must be even, so the last digit can be 0 or 2. Therefore, we have 2 choices for the last digit.
- After selecting the first and last digits, we are left with 2 digits from the remaining available digits. So, for the second digit, we have 2 choices, and for the third digit, we have 1 choice.
Thus, the total number of possible four-digit even numbers is:
3 × 2 × 2 × 1 = 6
So, the correct answer is 6.