In how many ways 7 members forming a committee out of 11 be selected so that 3 particular mimbers must be included ?

Options

A. 81
B. 70
C. 95
D. 125

Answer

To solve the problem of selecting a committee of 7 members from a total of 11 members, with the condition that 3 particular members must be included, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the total members and the requirement**: We have a total of 11 members and we need to form a committee of 7 members, ensuring that 3 specific members are included in the committee.

2. **Account for the members already included**: Since 3 particular members must be included, we can consider these members as already selected. This means we only need to select the remaining members.

3. **Calculate the remaining members**: After including the 3 particular members, we need to select 4 more members to complete the committee of 7. Therefore, we need to choose 4 members from the remaining members.

4. **Determine the number of remaining members**: Since we started with 11 members and have already included 3, we have \(11 - 3 = 8\) members left to choose from.

5. **Set up the combination formula**: We need to calculate the number of ways to choose 4 members from these 8 remaining members. This can be expressed using the combination formula \(nCr\), which is given by:
\[ nCr = \frac{n!}{r!(n-r)!} \]
Here, \(n = 8\) and \(r = 4\).

6. **Calculate \(8C4\)**:
\[ 8C4 = \frac{8!}{4!(8-4)!} = \frac{8!}{4! \cdot 4!} \]

7. **Simplify the factorials**:
\[ 8C4 = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} \]
Here, \(4!\) in the denominator cancels out part of \(8!\).

8. **Perform the calculations**:
- Calculate the numerator: \(8 \times 7 = 56\), \(56 \times 6 = 336\), \(336 \times 5 = 1680\).
- Calculate the denominator: \(4 \times 3 = 12\), \(12 \times 2 = 24\), \(24 \times 1 = 24\).
- Now divide:
\[ \frac{1680}{24} = 70 \]

9. **Conclusion**: Therefore, the total number of ways to select the committee of 7 members, including the 3 particular members, is **70**.