12 distinguishable balls are to be put in 4 identical boxes so that no box is empty the probability that one box contain 4 balls is...

To solve the problem of finding the probability that one box contains 4 balls when distributing 12 distinguishable balls into 4 identical boxes with the condition that no box is empty, we need to break it down into manageable steps. Let's start by determining the total number of ways to distribute the balls and then focus on the specific scenario where one box holds 4 balls.

Total Distributions of Balls

First, we need to calculate the total number of ways to distribute 12 distinguishable balls into 4 identical boxes, ensuring that no box is empty. This can be approached using the concept of Stirling numbers of the second kind, which count the ways to partition a set into non-empty subsets.

Using Stirling Numbers

The Stirling number of the second kind, denoted as S(n, k), gives the number of ways to partition n distinguishable objects into k non-empty indistinguishable subsets. For our case, we want S(12, 4).

Calculating S(12, 4) can be complex, but it can be derived using the recursive formula:

• S(n, k) = k * S(n-1, k) + S(n-1, k-1)

Using this recursive relation, we can compute S(12, 4) or refer to a table of Stirling numbers to find that S(12, 4) = 1, 034, 000.

Specific Distribution Scenario

Next, we need to find the number of ways to distribute the balls such that one box contains exactly 4 balls. We can break this down into two parts:

• Selecting 4 balls to go into one box.
• Distributing the remaining 8 balls into the other 3 boxes, ensuring that none of these boxes are empty.

Selecting 4 Balls

The number of ways to choose 4 balls from 12 is given by the combination formula:

• C(12, 4) = 12! / (4!(12-4)!) = 495.

Distributing Remaining Balls

Now, we need to distribute the remaining 8 balls into 3 boxes, ensuring that no box is empty. This can again be calculated using Stirling numbers:

• We need S(8, 3), which counts the ways to partition 8 balls into 3 non-empty boxes. From tables or calculations, we find that S(8, 3) = 1, 050.

Calculating the Desired Probability

Now that we have both components, we can find the total number of favorable outcomes:

• Total favorable outcomes = C(12, 4) * S(8, 3) = 495 * 1,050 = 519,750.

Finally, we can find the probability that one box contains exactly 4 balls by dividing the number of favorable outcomes by the total distributions:

• Probability = (Total favorable outcomes) / (Total distributions) = 519,750 / 1,034,000.

Calculating this gives us:

• Probability ≈ 0.502.

Final Thoughts

Thus, the probability that one box contains exactly 4 balls when distributing 12 distinguishable balls into 4 identical boxes, with no box being empty, is approximately 0.502. This means there is about a 50.2% chance of this specific distribution occurring. Understanding these combinatorial principles can be quite useful in various fields, including statistics, computer science, and operations research.