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 7.

Options

A. 31824
B. 1200
C. 9600
D. 15912

Explanation

To solve the problem of forming a committee of 7 members from a group of 10 men and 8 women, we can follow these steps:

### Step 1: Determine the total number of people
We have:
- 10 men
- 8 women

Total number of people = 10 + 8 = 18

### Step 2: Identify the committee size
We need to form a committee of 7 members.

### Step 3: Use the combination formula
The number of ways to choose \( r \) members from \( n \) members is given by the combination formula:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]
In our case, \( n = 18 \) and \( r = 7 \).

### Step 4: Substitute the values into the formula
Using the combination formula:
\[ C(18, 7) = \frac{18!}{7!(18-7)!} = \frac{18!}{7! \cdot 11!} \]

### Step 5: Simplify the factorial expression
We can simplify \( 18! \) as follows:
\[ C(18, 7) = \frac{18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12 \times 11!}{7! \cdot 11!} \]
The \( 11! \) cancels out:
\[ C(18, 7) = \frac{18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12}{7!} \]

### Step 6: Calculate \( 7! \)
Now calculate \( 7! \):
\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]

### Step 7: Calculate the numerator
Now we calculate the numerator:
\[ 18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12 \]
Calculating step by step:
- \( 18 \times 17 = 306 \)
- \( 306 \times 16 = 4896 \)
- \( 4896 \times 15 = 73440 \)
- \( 73440 \times 14 = 1028160 \)
- \( 1028160 \times 13 = 13365980 \)
- \( 13365980 \times 12 = 160391760 \)

### Step 8: Divide by \( 7! \)
Now divide the result by \( 7! \):
\[ C(18, 7) = \frac{160391760}{5040} = 31824 \]

### Final Answer
Thus, the number of different ways to form a committee of 7 members from 18 people is \( 31,824 \).

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