2026-04-01

Problem — 2026-04-01

Medium

Mediumcombinatorics

In how many ways can $5$ distinct balls be placed into $3$ distinct boxes so that no box is empty?

Enter an integer

Show solution

Total placements:

3^5 = 243

. Subtract placements using at most

2

boxes:

\binom{3}{1} \cdot 2^5 = 96

. Add back placements using exactly

1

box:

\binom{3}{2} \cdot 1^5 = 3

. By inclusion-exclusion:

243 - 96 + 3 = 150