Seven identical balls are to be placed into four distinct boxes so that no box remains empty. How many ways are possible?

Correct answer: 20

Let xi be the number of identical balls placed in box i. The conditions are x1 + x2 + x3 + x4 = 7 and xi ≥ 1 for every box. First place one ball in each of the four distinct boxes; this uses 4 balls and guarantees that no box is empty. Three identical balls remain to be distributed among four boxes, now allowing zero additional balls in a box. By stars and bars, the number of distributions is C(3 + 4 − 1, 4 − 1) = C(6,3) = 20. Therefore option D is correct. Treating the balls as distinct would greatly overcount, and allowing empty boxes would count a different problem.