The number of ways in which 9 identical balls can be placed in three identical boxes is

Options

A. 55
B. `(9!)/((3!)^(4))`
C. `(9!)/((3!^(3))`
D. `12`

Answer

To find the number of ways to place 9 identical balls into 3 identical boxes, we need to determine the different combinations of distributing the balls such that the sum of balls in the boxes equals 9. Since the boxes are identical, the order of distribution does not matter.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the number of non-negative integer solutions to the equation:
\[ x_1 + x_2 + x_3 = 9 \]
where \(x_1\), \(x_2\), and \(x_3\) represent the number of balls in each box.

2. **Identifying the Combinations**:
Since the boxes are identical, we will list the combinations of \(x_1\), \(x_2\), and \(x_3\) such that \(x_1 \geq x_2 \geq x_3\) to avoid counting permutations of the same distribution.

3. **Finding Valid Triplets**:
We will find the triplets \((x_1, x_2, x_3)\) that satisfy the equation \(x_1 + x_2 + x_3 = 9\):
- Start with the maximum number of balls in one box and decrease:
- (9, 0, 0)
- (8, 1, 0)
- (7, 2, 0)
- (7, 1, 1)
- (6, 3, 0)
- (6, 2, 1)
- (5, 4, 0)
- (5, 3, 1)
- (5, 2, 2)
- (4, 4, 1)
- (4, 3, 2)
- (3, 3, 3)

4. **Counting the Valid Combinations**:
Now we count the valid triplets we have found:
- (9, 0, 0)
- (8, 1, 0)
- (7, 2, 0)
- (7, 1, 1)
- (6, 3, 0)
- (6, 2, 1)
- (5, 4, 0)
- (5, 3, 1)
- (5, 2, 2)
- (4, 4, 1)
- (4, 3, 2)
- (3, 3, 3)

This gives us a total of 12 distinct combinations.

5. **Final Answer**:
Therefore, the total number of ways to place 9 identical balls into 3 identical boxes is:
\[ \boxed{12} \]