In how many different ways can it be done ?
10 men and 8 women out of which 5men are teachers, 3 men doctors and 2 businessmen ,Among the women , 3 are teachers, 2 doctors , 2 researches and 1 social worker.
A committee of 5 in which 2 men teachers , 2 women teachers and 1 doctor are there

Options

A. 75
B. 150
C. 214
D. 20

Explanation

To solve the problem of forming a committee of 5 members consisting of 2 male teachers, 2 female teachers, and 1 doctor, we can break down the solution into clear steps.

### Step 1: Identify the Groups
We have:
- **Men**: 10 total (5 teachers, 3 doctors, 2 businessmen)
- **Women**: 8 total (3 teachers, 2 doctors, 2 researchers, 1 social worker)

### Step 2: Select Male Teachers
We need to select 2 male teachers from the 5 available male teachers. The number of ways to choose 2 from 5 is calculated using the combination formula:

\[ \text{Number of ways to choose 2 male teachers} = \binom{5}{2} \]

Calculating this:

\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 3: Select Female Teachers
Next, we need to select 2 female teachers from the 3 available female teachers. Again, we use the combination formula:

\[ \text{Number of ways to choose 2 female teachers} = \binom{3}{2} \]

Calculating this:

\[ \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2}{2 \times 1} = 3 \]

### Step 4: Select a Doctor
We need to select 1 doctor from the total of 5 doctors (3 male doctors and 2 female doctors). The number of ways to choose 1 doctor is:

\[ \text{Number of ways to choose 1 doctor} = \binom{5}{1} \]

Calculating this:

\[ \binom{5}{1} = 5 \]

### Step 5: Calculate Total Combinations
Now that we have the number of ways to choose each group, we multiply these numbers together to find the total number of ways to form the committee:

\[ \text{Total ways} = \binom{5}{2} \times \binom{3}{2} \times \binom{5}{1} \]

Substituting the values we calculated:

\[ \text{Total ways} = 10 \times 3 \times 5 = 150 \]

### Final Answer
Thus, the total number of different ways to form this committee is **150**.

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