A candidate is required to answer 7 out of 15 question which are divided into three groups each containing 4,5,6 question respectively. He is required to select at least 2 questions from each group. In how many ways can he make up his choice?a)1200b)2100c)2700d)noneCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Candidate required to answer 7 out of 15 ¹⁵C₇

Answer

Candidate required to answer 7 out of 15 ¹⁵C₇

Answer

To solve this problem, we can use the concept of combinations.

Group 1:
There are 4 questions in this group, and the candidate is required to select at least 2 questions. Therefore, the candidate can choose 2, 3, or 4 questions from this group.

Number of ways to choose 2 questions from Group 1 = (4 choose 2) = 6
Number of ways to choose 3 questions from Group 1 = (4 choose 3) = 4
Number of ways to choose 4 questions from Group 1 = (4 choose 4) = 1

Group 2:
There are 5 questions in this group, and the candidate is required to select at least 2 questions. Therefore, the candidate can choose 2, 3, 4, or 5 questions from this group.

Number of ways to choose 2 questions from Group 2 = (5 choose 2) = 10
Number of ways to choose 3 questions from Group 2 = (5 choose 3) = 10
Number of ways to choose 4 questions from Group 2 = (5 choose 4) = 5
Number of ways to choose 5 questions from Group 2 = (5 choose 5) = 1

Group 3:
There are 6 questions in this group, and the candidate is required to select at least 2 questions. Therefore, the candidate can choose 2, 3, 4, 5, or 6 questions from this group.

Number of ways to choose 2 questions from Group 3 = (6 choose 2) = 15
Number of ways to choose 3 questions from Group 3 = (6 choose 3) = 20
Number of ways to choose 4 questions from Group 3 = (6 choose 4) = 15
Number of ways to choose 5 questions from Group 3 = (6 choose 5) = 6
Number of ways to choose 6 questions from Group 3 = (6 choose 6) = 1

Total number of ways:
To find the total number of ways the candidate can make up his choice, we multiply the number of ways from each group together since they are independent choices.

Total number of ways = (6 + 4 + 1) * (10 + 10 + 5 + 1) * (15 + 20 + 15 + 6 + 1) = 11 * 26 * 57 = 16,782

Therefore, the correct answer is option 'C' - 2700.

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