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 1: Understand the problem
We need to choose 5 red balls from a total of 10 red balls and 4 white balls from a total of 8 white balls.

### Step 2: Use the combination formula
The number of ways to choose r items from n items is given by the combination formula:
\[ nC_r = \frac{n!}{r!(n - r)!} \]
Where \( n! \) (n factorial) is the product of all positive integers up to n.

### Step 3: Calculate the number of ways to choose red balls
We need to choose 5 red balls from 10 red balls:
\[ 10C_5 = \frac{10!}{5!(10 - 5)!} = \frac{10!}{5!5!} \]

Calculating \( 10C_5 \):
\[ 10C_5 = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = \frac{30240}{120} = 252 \]

### Step 4: Calculate the number of ways to choose white balls
Now, we need to choose 4 white balls from 8 white balls:
\[ 8C_4 = \frac{8!}{4!(8 - 4)!} = \frac{8!}{4!4!} \]

Calculating \( 8C_4 \):
\[ 8C_4 = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = \frac{1680}{24} = 70 \]

### Step 5: Calculate the total number of ways
To find the total number of ways to draw 5 red and 4 white balls, we multiply the number of ways to choose the red balls by the number of ways to choose the white balls:
\[ \text{Total ways} = 10C_5 \times 8C_4 = 252 \times 70 \]

Calculating the total:
\[ \text{Total ways} = 17640 \]

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

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