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

We are given 12 identical balls to be placed in 3 identical boxes. We need to find the probability that at least one box contains exactly 3 balls. Since the boxes are identical, the distributions are unordered, meaning that the order of the boxes does not matter. Therefore, we consider the partitions of the number 12 into up to 3 parts, where each part represents the number of balls in a box, and the parts are in non-decreasing order to avoid counting permutations of the same distribution multiple times.

First, we find the total number of ways to distribute the balls. This is equivalent to the number of partitions of 12 into at most 3 parts. We list all possible partitions:

• (0, 0, 12)
• (0, 1, 11)
• (0, 2, 10)
• (0, 3, 9)
• (0, 4, 8)
• (0, 5, 7)
• (0, 6, 6)
• (1, 1, 10)
• (1, 2, 9)
• (1, 3, 8)
• (1, 4, 7)
• (1, 5, 6)
• (2, 2, 8)
• (2, 3, 7)
• (2, 4, 6)
• (2, 5, 5)
• (3, 3, 6)
• (3, 4, 5)
• (4, 4, 4)

Counting these, we have 19 distinct distributions. Therefore, the total number of possible outcomes is 19.

Next, we find the favorable outcomes where at least one box has exactly 3 balls. We look for partitions that include the number 3:

• (0, 3, 9) — contains one 3
• (1, 3, 8) — contains one 3
• (2, 3, 7) — contains one 3
• (3, 3, 6) — contains two 3's
• (3, 4, 5) — contains one 3

We have 5 such distributions. Note that the partition (3, 3, 6) is included because it has at least one box with exactly 3 balls.

The number of favorable outcomes is 5.

Probability is defined as the ratio of the number of favorable outcomes to the total number of outcomes. Therefore, the probability is:

$$ \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{5}{19} $$

Hence, the correct answer is Option B.