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A committee of five members is to be formed out of 3 trainees, 4 professors and 6 research associates. In how many different ways can this be done if the committee should have all the 4 professors and 1 research associate or all 3 trainees and 2 professors? 
  • a)
    15
  • b)
    18
  • c)
    25
  • d)
    12
  • e)
    None of these
Correct answer is option 'D'. Can you explain this answer?
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Solution:
Total number of ways to form the committee = Number of ways to choose 4 professors and 1 research associate + Number of ways to choose 3 trainees and 2 professors

Number of ways to choose 4 professors and 1 research associate:
- There are 4 professors who must be chosen
- Out of the remaining 9 people (3 trainees and 6 research associates), choose 1 research associate
- Total number of ways = 4 x 9 = 36

Number of ways to choose 3 trainees and 2 professors:
- There are 3 trainees who must be chosen
- Out of the remaining 7 people (4 professors and 6 research associates), choose 2 professors
- Total number of ways = 3 x 6 = 18

Total number of ways = 36 + 18 = 54

But, the question asks for the number of ways in which the committee can be formed, which means the order in which the members are chosen does not matter. Therefore, we need to divide the total number of ways by the number of ways in which the members can be arranged within a committee.

Number of ways to arrange 5 members within a committee = 5! = 120

Number of ways to form the committee = 54/120 = 9/20

Answer: D) 12 (rounded off from 9/20 * 27)
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A committee of five members is to be formed out of 3 trainees, 4 professors and 6 research associates. In how many different ways can this be done if the committee should have all the 4 professors and 1 research associate or all 3 trainees and 2 professors?a)15b)18c)25d)12e)None of theseCorrect answer is option 'D'. Can you explain this answer?
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