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. Assuming that balls of the same colour are distinguishable is

Options

A. 40
B. 60
C. 180
D. 2000

Answer

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

### Step 1: Calculate the number of ways to select 3 red balls from 6 red balls.
We use the combination formula \( nCr \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Here, \( n = 6 \) (red balls) and \( r = 3 \) (we want to select 3 red balls).

\[ \text{Number of ways} = \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6!}{3! \cdot 3!} \]

Calculating this:

\[ = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = \frac{120}{6} = 20 \]

### Step 2: Calculate the number of ways to select 3 white balls from 5 white balls.
Similarly, for white balls, we have \( n = 5 \) and \( r = 3 \).

\[ \text{Number of ways} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \cdot 2!} \]

Calculating this:

\[ = \frac{5 \times 4}{2 \times 1} = \frac{20}{2} = 10 \]

### Step 3: Calculate the number of ways to select 3 blue balls from 5 blue balls.
For blue balls, we also have \( n = 5 \) and \( r = 3 \).

\[ \text{Number of ways} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \cdot 2!} \]

Calculating this:

\[ = \frac{5 \times 4}{2 \times 1} = \frac{20}{2} = 10 \]

### Step 4: Calculate the total number of ways to select 3 balls of each color.
Now, we multiply the number of ways to select red, white, and blue balls:

\[ \text{Total ways} = (\text{Ways to select red}) \times (\text{Ways to select white}) \times (\text{Ways to select blue}) \]

\[ = 20 \times 10 \times 10 = 2000 \]

### Final Answer
Thus, the total number of ways to select 9 balls consisting of 3 red, 3 white, and 3 blue balls is **2000**.

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