If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is :

Options

A. `220((1)/(3))^(12)`
B. `22((1)/(3))^(11)`
C. `55((2)/(3))^(11)`
D. `55((2)/(3))^(10)`

Answer

To solve the problem of finding the probability that one of the boxes contains exactly 3 balls when distributing 12 identical balls into 3 identical boxes, we can follow these steps:

### Step 1: Determine the total number of ways to distribute 12 identical balls into 3 identical boxes.

Since the boxes are identical, we can use the "stars and bars" theorem. The total number of distributions of \( n \) identical items into \( r \) identical boxes is given by the number of partitions of \( n \) into at most \( r \) parts.

For 12 balls and 3 boxes, we need to find the partitions of 12 into at most 3 parts. The possible distributions are:
- (12, 0, 0)
- (11, 1, 0)
- (10, 2, 0)
- (10, 1, 1)
- (9, 3, 0)
- (9, 2, 1)
- (8, 4, 0)
- (8, 3, 1)
- (8, 2, 2)
- (7, 5, 0)
- (7, 4, 1)
- (7, 3, 2)
- (6, 6, 0)
- (6, 5, 1)
- (6, 4, 2)
- (6, 3, 3)
- (5, 5, 2)
- (5, 4, 3)
- (4, 4, 4)

Counting these partitions gives us a total of 15 ways to distribute 12 balls into 3 identical boxes.

### Step 2: Determine the number of favorable outcomes where one box contains exactly 3 balls.

If one box contains exactly 3 balls, we need to distribute the remaining 9 balls into the other 2 boxes. The number of ways to distribute 9 identical balls into 2 identical boxes can also be found using the partition method.

The possible distributions of 9 balls into 2 boxes are:
- (9, 0)
- (8, 1)
- (7, 2)
- (6, 3)
- (5, 4)
- (4, 4)

Counting these distributions gives us a total of 6 ways to distribute 9 balls into 2 identical boxes.

### Step 3: Calculate the probability.

The probability \( P \) that one of the boxes contains exactly 3 balls is given by the ratio of the number of favorable outcomes to the total outcomes:

\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{6}{15} = \frac{2}{5} \]

### Conclusion

Thus, the probability that one of the boxes contains exactly 3 balls is \( \frac{2}{5} \).