To find the probability that one of the boxes contains exactly 3 balls when placing 12 identical balls into 3 identical boxes, we can follow these steps:
### Step-by-Step Solution:
1. **Understanding the Problem**: We need to calculate the probability that exactly one of the three identical boxes contains exactly 3 balls when distributing 12 identical balls among them.
2. **Identifying the Total Outcomes**: Since the boxes are identical, we need to find the total number of ways to distribute 12 identical balls into 3 identical boxes. The number of ways to partition 12 into up to 3 parts can be calculated using the partition function or generating functions, but for simplicity, we will consider the successful outcomes directly.
3. **Successful Outcomes**: We want to find the number of distributions where one box has exactly 3 balls. This means we need to distribute the remaining 9 balls into the 3 boxes, where one box already has 3 balls.
4. **Using Stars and Bars Method**: We can represent the remaining 9 balls as stars and use bars to separate the boxes. The formula for distributing \( n \) identical items into \( r \) distinct groups is given by:
\[
\text{Number of ways} = \binom{n + r - 1}{r - 1}
\]
In our case, \( n = 9 \) (remaining balls) and \( r = 3 \) (boxes):
\[
\text{Number of ways} = \binom{9 + 3 - 1}{3 - 1} = \binom{11}{2}
\]
5. **Calculating \(\binom{11}{2}\)**:
\[
\binom{11}{2} = \frac{11 \times 10}{2 \times 1} = 55
\]
6. **Total Outcomes**: The total number of ways to distribute 12 identical balls into 3 identical boxes can be calculated as the number of partitions of 12 into at most 3 parts. The partitions of 12 into at most 3 parts are:
- (12)
- (11, 1)
- (10, 2)
- (9, 3)
- (8, 4)
- (8, 3, 1)
- (7, 5)
- (7, 4, 1)
- (7, 3, 2)
- (6, 6)
- (6, 5, 1)
- (6, 4, 2)
- (6, 3, 3)
- (5, 5, 2)
- (5, 4, 3)
- (4, 4, 4)
Counting these, we find there are 15 distinct partitions.
7. **Calculating Probability**: The probability \( P \) that one of the boxes contains exactly 3 balls is given by:
\[
P = \frac{\text{Number of successful outcomes}}{\text{Total outcomes}} = \frac{55}{15}
\]
8. **Simplifying the Probability**: We can simplify this fraction:
\[
P = \frac{11}{3}
\]
9. **Final Probability**: The probability that one of the boxes contains exactly 3 balls is:
\[
P = \frac{55}{3} \times \left(\frac{2}{3}\right)^{11}
\]
