Out of 10 red and 8 white balls , 5 red and 4 white balls can be drawn in number of ways

Options

A. `""^8 C_5 times "^10 C_4`
B. `""^10 C_5 times "^8C_4`
C. `"^18C_8`
D. none of these

Answer

To solve the problem of drawing 5 red balls from 10 red balls and 4 white balls from 8 white balls, we will use the concept of combinations. The formula for combinations is given by:

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

where \( n \) is the total number of items, \( r \) is the number of items to choose, and \( ! \) denotes factorial.

### Step 1: Calculate the number of ways to choose 5 red balls from 10 red balls.

Using the combination formula:

\[ ^{10}C_5 = \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!} \]

Calculating \( 10! \), \( 5! \):

\[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5! \]

Thus,

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

### Step 2: Calculate the number of ways to choose 4 white balls from 8 white balls.

Using the combination formula again:

\[ ^{8}C_4 = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} \]

Calculating \( 8! \):

\[ 8! = 8 \times 7 \times 6 \times 5 \times 4! \]

Thus,

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

### Step 3: Calculate the total number of ways to choose 5 red and 4 white balls.

Since the selections of red and white balls are independent, we multiply the number of ways to choose red balls by the number of ways to choose white balls:

\[ \text{Total ways} = ^{10}C_5 \times ^{8}C_4 = 252 \times 70 \]

Calculating this gives:

\[ 252 \times 70 = 17640 \]

### Final Answer:

The total number of ways to draw 5 red balls and 4 white balls is **17640**.

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