A committee of 5 members is going to be formed from 3 trainees, 4 professors and 6 research associates. How many ways can they be selected, if in committee, there are 2 trainees and 3 research associates?

Options

A. a )15
B. b ) 45
C. c )60
D. d)9

Answer

To solve the problem of forming a committee of 5 members consisting of 2 trainees and 3 research associates from a pool of 3 trainees, 4 professors, and 6 research associates, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Groups**:
- We have 3 trainees, 4 professors, and 6 research associates.
- We need to select 2 trainees and 3 research associates.

2. **Calculate the Number of Ways to Choose Trainees**:
- We need to select 2 trainees from the 3 available trainees.
- The number of ways to choose 2 trainees from 3 can be calculated using the combination formula:
\[ \text{Number of ways to choose 2 trainees} = \binom{3}{2} \]
- This can be calculated as:
\[ \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2 \times 1}{2 \times 1 \times 1} = 3 \]

3. **Calculate the Number of Ways to Choose Research Associates**:
- We need to select 3 research associates from the 6 available research associates.
- The number of ways to choose 3 research associates from 6 can be calculated using the combination formula:
\[ \text{Number of ways to choose 3 research associates} = \binom{6}{3} \]
- This can be calculated as:
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

4. **Calculate the Total Number of Ways to Form the Committee**:
- Since the selections of trainees and research associates are independent, we multiply the number of ways to choose the trainees by the number of ways to choose the research associates:
\[ \text{Total ways} = \binom{3}{2} \times \binom{6}{3} = 3 \times 20 = 60 \]

5. **Conclusion**:
- Therefore, the total number of ways to form the committee of 5 members with 2 trainees and 3 research associates is **60**.