In how many ways can 15 billiard balls be arranged in a row if 3 are red, 4 are white and 8 are black?

Options

A. 12
B. 18
C. 96
D. none of these

Answer

To solve the problem of arranging 15 billiard balls in a row, where there are 3 red balls, 4 white balls, and 8 black balls, we can use the formula for permutations of multiset. The formula for the number of arrangements of n items where there are groups of indistinguishable items is given by:

\[ \frac{n!}{n_1! \cdot n_2! \cdot n_3!} \]

Where:
- \( n \) is the total number of items,
- \( n_1, n_2, n_3 \) are the counts of indistinguishable items.

### Step-by-step Solution:

1. **Identify the Total Number of Balls**:
- We have a total of 15 billiard balls: 3 red, 4 white, and 8 black.

2. **Write Down the Factorial of the Total Balls**:
- The total number of arrangements is calculated using \( 15! \) since we have 15 balls.

3. **Write Down the Factorials of Each Color**:
- For the red balls: \( 3! \)
- For the white balls: \( 4! \)
- For the black balls: \( 8! \)

4. **Set Up the Formula**:
- Using the formula for permutations of a multiset, we have:
\[ \text{Number of arrangements} = \frac{15!}{3! \cdot 4! \cdot 8!} \]

5. **Calculate Each Factorial**:
- \( 15! = 1307674368000 \)
- \( 3! = 6 \)
- \( 4! = 24 \)
- \( 8! = 40320 \)

6. **Substitute the Values into the Formula**:
\[ \text{Number of arrangements} = \frac{1307674368000}{6 \cdot 24 \cdot 40320} \]

7. **Calculate the Denominator**:
- First, calculate \( 6 \cdot 24 = 144 \).
- Then, calculate \( 144 \cdot 40320 = 5806080 \).

8. **Final Calculation**:
\[ \text{Number of arrangements} = \frac{1307674368000}{5806080} = 225792 \]

Therefore, the total number of ways to arrange the 15 billiard balls is **225792**.