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 there are 4 professors and 1 research associate or 3 trainees and 2 professors?

Options

A. 12
B. 13
C. 24
D. 52

Answer

To solve the problem of forming a committee of 5 members from 3 trainees, 4 professors, and 6 research associates, we will consider the two scenarios given in the question:

1. **Scenario 1**: 4 professors and 1 research associate.
2. **Scenario 2**: 3 trainees and 2 professors.

We will calculate the number of ways to form the committee for each scenario and then add the results together.

### Step-by-Step Solution:

**Step 1: Calculate the number of ways for Scenario 1 (4 professors and 1 research associate)**

- We need to select 4 professors from the 4 available professors. The number of ways to do this is given by the combination formula \( \binom{n}{r} \), which represents the number of ways to choose \( r \) items from \( n \) items without regard to the order of selection.

\[ \text{Number of ways to select 4 professors} = \binom{4}{4} = 1 \]

- Next, we need to select 1 research associate from the 6 available research associates.

\[ \text{Number of ways to select 1 research associate} = \binom{6}{1} = 6 \]

- Therefore, the total number of ways for Scenario 1 is:

\[ \text{Total for Scenario 1} = \binom{4}{4} \times \binom{6}{1} = 1 \times 6 = 6 \]

**Step 2: Calculate the number of ways for Scenario 2 (3 trainees and 2 professors)**

- We need to select all 3 trainees from the 3 available trainees.

\[ \text{Number of ways to select 3 trainees} = \binom{3}{3} = 1 \]

- Next, we need to select 2 professors from the 4 available professors.

\[ \text{Number of ways to select 2 professors} = \binom{4}{2} = 6 \]

- Therefore, the total number of ways for Scenario 2 is:

\[ \text{Total for Scenario 2} = \binom{3}{3} \times \binom{4}{2} = 1 \times 6 = 6 \]

**Step 3: Combine the results from both scenarios**

- Now we add the totals from Scenario 1 and Scenario 2 to get the final answer:

\[ \text{Total ways to form the committee} = \text{Total for Scenario 1} + \text{Total for Scenario 2} = 6 + 6 = 12 \]

### Final Answer:
The total number of ways to form the committee is **12**.

---