(a) In how many ways can you distribute 4 distinguishable balls among 3 indistinguishable boxes?

(b) In how many ways can you distribute 4 indistinguishable balls among 5 distinguishable boxes?

(a)

This time we don't care about what balls are in what box, but the groupings of the balls a, b, c, and d.

abcd (1 case)

abc, d (4 different cases)

ab, cd (4 choose 2 / 2 = 3 cases)

ab, c, d (4 choose 2 = 6 cases)

That's all: 1 + 4 + 3 + 6 = 14 total cases.