In how many ways can 5 red and 4 white balls be drawn from a bag containing 10 red and 8 white balls

Options

A. `""^(8)C_(5)xx""^(10)C_(4)`
B. `""^(10)C_(5)xx""^(8)C_(4)`
C. `""^(18)C_(9)`
D. none of these

Answer

To solve the problem of how many ways we can draw 5 red and 4 white balls from a bag containing 10 red and 8 white balls, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the total number of balls**:
- We have 10 red balls and 8 white balls in the bag.

2. **Determine the number of balls to be drawn**:
- We need to draw 5 red balls and 4 white balls.

3. **Calculate the combinations for red balls**:
- The number of ways to choose 5 red balls from 10 can be calculated using the combination formula:
\[ \text{Number of ways to choose 5 red balls} = \binom{10}{5} \]
- The formula for combinations is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
- Applying this for our case:
\[ \binom{10}{5} = \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252 \]

4. **Calculate the combinations for white balls**:
- The number of ways to choose 4 white balls from 8 can be calculated similarly:
\[ \text{Number of ways to choose 4 white balls} = \binom{8}{4} \]
- Applying the combination formula:
\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

5. **Multiply the combinations**:
- Since the selection of red and white balls are independent events, we multiply the number of combinations:
\[ \text{Total ways} = \binom{10}{5} \times \binom{8}{4} = 252 \times 70 \]
- Calculating this gives:
\[ 252 \times 70 = 17640 \]

### Final Answer:
The total number of ways to draw 5 red and 4 white balls from the bag is **17640**.