In how many ways 4 boys can be chosen from 7 boys to make a committee?

Answer

To solve the problem of how many ways 4 boys can be chosen from 7 boys to make a committee, we will use the concept of combinations.

### Step-by-Step Solution:

1. **Identify the values of n and r**:
- Here, \( n \) is the total number of boys, which is 7.
- \( r \) is the number of boys to be chosen, which is 4.

2. **Use the combination formula**:
- The formula for combinations is given by:
\[ \binom{n}{r} = \frac{n!}{(n-r)! \cdot r!} \]
- In our case, we need to calculate \( \binom{7}{4} \).

3. **Substitute the values into the formula**:
- Substitute \( n = 7 \) and \( r = 4 \) into the formula:
\[ \binom{7}{4} = \frac{7!}{(7-4)! \cdot 4!} = \frac{7!}{3! \cdot 4!} \]

4. **Calculate the factorials**:
- Calculate \( 7! \), \( 3! \), and \( 4! \):
- \( 7! = 7 \times 6 \times 5 \times 4! \)
- \( 3! = 3 \times 2 \times 1 = 6 \)
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

5. **Substitute the factorials back into the equation**:
- Now substitute these values back into the combination formula:
\[ \binom{7}{4} = \frac{7 \times 6 \times 5 \times 4!}{3! \cdot 4!} \]
- The \( 4! \) in the numerator and denominator cancels out:
\[ \binom{7}{4} = \frac{7 \times 6 \times 5}{3!} = \frac{7 \times 6 \times 5}{6} \]

6. **Simplify the expression**:
- Now simplify the expression:
\[ \binom{7}{4} = 7 \times 5 = 35 \]

7. **Conclusion**:
- Therefore, the number of ways to choose 4 boys from 7 boys to form a committee is **35**.