(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.
