The number of ways of distributing 12 identical balls in 5 different boxes so that none of the box is empty is

Options

A. `.^(12)C_(5)`
B. `.^(17)C_(5)`
C. `.^(16)C_(4)`
D. `.^(11)C_(4)`

Answer

To solve the problem of distributing 12 identical balls into 5 different boxes such that none of the boxes is empty, we can use the combinatorial method known as the "stars and bars" theorem. However, since we need to ensure that no box is empty, we will first place one ball in each box and then distribute the remaining balls.

### Step-by-Step Solution:

1. **Initial Distribution**:
Since we want to ensure that none of the boxes is empty, we start by placing one ball in each of the 5 boxes. This uses up 5 balls (1 in each box).

\[ \text{Balls used} = 5 \]

\[ \text{Remaining balls} = 12 - 5 = 7 \]

2. **Distributing Remaining Balls**:
Now, we need to distribute the remaining 7 identical balls into the 5 boxes. Since the boxes can now be empty (as we have already ensured that each has at least one ball), we can use the stars and bars method.

3. **Applying the Stars and Bars Theorem**:
The number of ways to distribute \( n \) identical items (balls) into \( r \) distinct groups (boxes) is given by the formula:

\[ \binom{n + r - 1}{r - 1} \]

Here, \( n \) is the number of remaining balls (7), and \( r \) is the number of boxes (5).

\[ \text{Using the formula: } \binom{7 + 5 - 1}{5 - 1} = \binom{11}{4} \]

4. **Calculating the Binomial Coefficient**:
Now, we need to calculate \( \binom{11}{4} \):

\[ \binom{11}{4} = \frac{11!}{4!(11 - 4)!} = \frac{11!}{4! \cdot 7!} \]

This can be simplified as follows:

\[ = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} \]

Calculating the numerator:

\[ 11 \times 10 = 110 \]
\[ 110 \times 9 = 990 \]
\[ 990 \times 8 = 7920 \]

Now calculating the denominator:

\[ 4 \times 3 = 12 \]
\[ 12 \times 2 = 24 \]
\[ 24 \times 1 = 24 \]

Now, divide the numerator by the denominator:

\[ \frac{7920}{24} = 330 \]

5. **Final Answer**:
Therefore, the total number of ways to distribute 12 identical balls into 5 different boxes such that none of the boxes is empty is:

\[ \boxed{330} \]