Five balls covering four bins

Question

Five balls are thrown independently into 4 bins, each ball equally likely to land in any bin. What is the probability that every bin receives at least one ball?

Accepted Answer

There are 4^5 = 1024 equally likely assignments. Apply inclusion-exclusion to the "empty bin" events, now with 4 bins. - All assignments: 4^5 = 1024. - Subtract those skipping a chosen bin: \binom{4}{1}\cdot 3^5 = 4 \cdot 243 = 972. - Add back those skipping a chosen pair: \binom{4}{2}\cdot 2^5 = 6 \cdot 32 = 192. - Subtract those skipping a chosen triple: \binom{4}{3}\cdot 1^5 = 4 \cdot 1 = 4. Assignments hitting all four bins: 1024 - 972 + 192 - 4 = 240. So P(	ext{all occupied}) = \frac{240}{1024} = \frac{15}{64} \approx 0.2344. With only one spare ball beyond the four bins, covering them …

Five balls covering four bins

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