Six balls, three bins, none empty

Question

Six balls are thrown independently into 3 bins, each ball equally likely to land in any bin. What is the probability that no bin is left empty?

Accepted Answer

There are 3^6 = 729 equally likely assignments. Count those where every bin is used with inclusion-exclusion on the "empty bin" events. - All assignments: 3^6 = 729. - Subtract those avoiding a chosen bin: 3 \cdot 2^6 = 3 \cdot 64 = 192. - Add back those avoiding a chosen pair of bins: 3 \cdot 1^6 = 3. Assignments with all bins occupied: 729 - 192 + 3 = 540. So P(	ext{all occupied}) = \frac{540}{729} = \frac{20}{27} \approx 0.7407. Adding one more ball (versus five balls) noticeably raises the chance every bin is hit. Takeaway. Surjection count \sum{k=0}^{n}(-1)^k\binom{n}{k}(n-k)^m over n^m…

Six balls, three bins, none empty

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