Question

A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. The number of ways in which he can choose the 7 questions is:

• A. 1272
• B. 780
• C. 640
• D. 820

Hint:

When solving problems involving combinations, break the problem down into cases and compute the number of ways for each case. Then, add the results to get the total number of possibilities.

Answer 1

The Correct Option is B

The total number of questions is 12, divided into two groups, each containing 6 questions. We are required to choose 7 questions with the constraint that no more than 5 questions can be selected from either group. Step 1: Identify the possible combinations. Since we are selecting 7 questions, we have the following two possible scenarios for choosing questions from each group: - Choose 5 questions from one group and 2 questions from the other group. - Choose 4 questions from one group and 3 questions from the other group. Step 2: Calculate the number of ways for each scenario. # Scenario 1: Choose 5 questions from one group and 2 questions from the other group. - The number of ways to choose 5 questions from a group of 6 questions is given by \( \binom{6}{5} \). - The number of ways to choose 2 questions from the other group of 6 questions is given by \( \binom{6}{2} \). Thus, the number of ways for this scenario is: \[ \binom{6}{5} \times \binom{6}{2} = 6 \times 15 = 90 \] # Scenario 2: Choose 4 questions from one group and 3 questions from the other group. - The number of ways to choose 4 questions from a group of 6 questions is given by \( \binom{6}{4} \). - The number of ways to choose 3 questions from the other group of 6 questions is given by \( \binom{6}{3} \). Thus, the number of ways for this scenario is: \[ \binom{6}{4} \times \binom{6}{3} = 15 \times 20 = 300 \] Step 3: Total number of ways. Since there are two groups, we can have the following combinations: - Choose 5 questions from Group 1 and 2 questions from Group 2: \( 90 \) - Choose 4 questions from Group 1 and 3 questions from Group 2: \( 300 \) Thus, the total number of ways to select 7 questions is: \[ 90 + 300 = 780 \] Therefore, the correct answer is option (B)