Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.

Options

A. 1000
B. 2000
C. 3000
D. 4000

Explanation

To solve the problem of selecting 9 balls from 6 red balls, 5 white balls, and 5 blue balls, with the condition that each selection consists of 3 balls of each color, we can follow these steps:

### Step 1: Understand the selection requirement
We need to select a total of 9 balls, with 3 balls from each color:
- 3 red balls
- 3 white balls
- 3 blue balls

### Step 2: Calculate the number of ways to select the balls of each color
We will use the combination formula to determine the number of ways to select the required number of balls from each color. The combination formula is given by:

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

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

### Step 3: Calculate combinations for each color
1. **Red Balls**: We need to select 3 balls from 6 red balls.
\[ C(6, 3) = \frac{6!}{3!(6 - 3)!} = \frac{6!}{3! \cdot 3!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

2. **White Balls**: We need to select 3 balls from 5 white balls.
\[ C(5, 3) = \frac{5!}{3!(5 - 3)!} = \frac{5!}{3! \cdot 2!} = \frac{5 \times 4}{2 \times 1} = 10 \]

3. **Blue Balls**: We need to select 3 balls from 5 blue balls.
\[ C(5, 3) = \frac{5!}{3!(5 - 3)!} = \frac{5!}{3! \cdot 2!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 4: Calculate the total number of ways to select the balls
The total number of ways to select the balls is the product of the combinations for each color:
\[ \text{Total Ways} = C(6, 3) \times C(5, 3) \times C(5, 3) = 20 \times 10 \times 10 \]

Calculating this gives:
\[ \text{Total Ways} = 20 \times 10 \times 10 = 2000 \]

### Conclusion
Thus, the total number of ways to select 9 balls such that there are 3 balls of each color is **2000**.

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