Is there a counting technique for this?
(a) In how many ways can you distribute 4 distinguishable balls among 6 indistinguishable boxes?
(b) In how many ways can you distribute 4 indistinguishable balls among 5 distinguishable boxes?
YES! There is a counting method oulined here in Wikipedia: https://en.wikipedia.org/wiki/Twelvefold_way
Look at the table at the very bottom of the page. EVERY case of distributions of balls into boxes (or whatever the case may be) is there with the CORRECT formula to be used. Good luck.