There are 25 students in a class . Find the number of ways in which a commitee of 3 students is to be formed.

Options

A. 2200
B. 2300
C. 2400
D. 3200

Answer

To find the number of ways to form a committee of 3 students from a class of 25 students, we can use the concept of combinations. The formula for combinations is given by:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

Where:
- \( n \) is the total number of items (students in this case),
- \( r \) is the number of items to choose (students to be selected for the committee),
- \( ! \) denotes factorial, which is the product of all positive integers up to that number.

### Step-by-step Solution:

1. **Identify the values of n and r**:
- Total students \( n = 25 \)
- Students to be selected for the committee \( r = 3 \)

2. **Apply the combination formula**:
\[ \binom{25}{3} = \frac{25!}{3!(25-3)!} = \frac{25!}{3! \cdot 22!} \]

3. **Simplify the factorials**:
- We can expand \( 25! \) as \( 25 \times 24 \times 23 \times 22! \).
- So, we have:
\[ \binom{25}{3} = \frac{25 \times 24 \times 23 \times 22!}{3! \cdot 22!} \]

4. **Cancel out \( 22! \)**:
\[ \binom{25}{3} = \frac{25 \times 24 \times 23}{3!} \]

5. **Calculate \( 3! \)**:
\[ 3! = 3 \times 2 \times 1 = 6 \]

6. **Substitute \( 3! \) back into the equation**:
\[ \binom{25}{3} = \frac{25 \times 24 \times 23}{6} \]

7. **Calculate the numerator**:
- First calculate \( 25 \times 24 = 600 \)
- Then calculate \( 600 \times 23 = 13800 \)

8. **Divide by \( 6 \)**:
\[ \binom{25}{3} = \frac{13800}{6} = 2300 \]

### Final Answer:
The number of ways to form a committee of 3 students from 25 students is **2300**.