PERMUTATIONS and COMBINATIONS!?

Question

In how many ways can 5 letters be mailed if there are 2 mailboxes available? if there are 4 mailboxes available? Please explain your way ans answer
Thanks

Answer 1

Part 1 is a combination problem with 6 ways to do it:
all 5 in mailbox A : 5C5 = 1 way
4 in mailbox A and 1 in mailbox B: 5C4 X 1C1 = 5 ways
3 in box A and 2 in B: 5C3 X 2C2 = 10 ways
etc.
.

Part 2 has many more ways obviously. You could put all 5 into box A, B, C, or D so that's 4 ways.
Also (listing boxes A - D)

4,1,0,0 which would be 5C4 X 1C1 then times 4!/2! for the different ways the 4,1,0,0 could be arranged

3,1,1,0 which is 5C3 X 2C1 X 1C1 then times 4!/2! again

same with
3,2,0,0
2,2,1,0
2,1,1,1