I'm assuming that we have no restrictions ( some boxes can be empty)
1. k distinct balls into n distinct boxes = n^k = 4^4 = 256 ways
2. k indentical balls into n identical boxes ....this one is tough....it involves partitions.....as I'm going to have to leave in a minute....I'll see if I can give you an answer later
3. k identical balls into n distinct boxes = C (k +n - 1 , n-1) = C( 4 + 4 -1, 4 -1) = C(7,4) = 35 ways
C (7,4) is the number of ways to choose 4 things from 7 ....in case you are not familiar with this notation