A box contains 7 red , 6 white and 4 blue balls. Number of ways of selection of three red balls is

Options

A. 35
B. 45
C. 27
D. 36

Answer

To solve the problem of selecting 3 red balls from a total of 7 red balls, we can use the concept of combinations. The formula for combinations is given by:

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

where:
- \( n \) is the total number of items (in this case, red balls),
- \( r \) is the number of items to choose,
- \( ! \) 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 red balls (n) = 7
- Balls to choose (r) = 3

2. **Apply the combination formula**:
\[ C(7, 3) = \frac{7!}{3!(7 - 3)!} = \frac{7!}{3! \cdot 4!} \]

3. **Expand the factorials**:
- \( 7! = 7 \times 6 \times 5 \times 4! \)
- Therefore, we can rewrite the combination as:
\[ C(7, 3) = \frac{7 \times 6 \times 5 \times 4!}{3! \cdot 4!} \]

4. **Cancel out the common factorial (4!)**:
\[ C(7, 3) = \frac{7 \times 6 \times 5}{3!} \]

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

6. **Substitute \( 3! \) back into the equation**:
\[ C(7, 3) = \frac{7 \times 6 \times 5}{6} \]

7. **Simplify the expression**:
- The 6 in the numerator and denominator cancels out:
\[ C(7, 3) = 7 \times 5 = 35 \]

8. **Conclusion**:
The number of ways to select 3 red balls from 7 red balls is **35**.