Find the number of ways to distribute 10 distinct balls into 3 distinct boxes such that no box is empty.

Solution

1. The total number of ways to distribute 10 distinct balls into 3 distinct boxes without restrictions is $3^{10}=59049$.
2. By the Principle of Inclusion-Exclusion, we subtract the cases where at least one box is empty: $3^{10}-\left(\genfrac{}{}{0ex}{}{3}{1}\right)2^{10}+\left(\genfrac{}{}{0ex}{}{3}{2}\right)1^{10}$.
3. Substituting the values, we get $59049 - 3(1024) + 3(1) = 59049 - 3072 + 3 = 55980$.
4. Hence the answer is (A).