Combinatorics - What is the number of ways to color n objects with 3 colors if every color must be used at least once?

problem1) What is the number of ways to color n objects with 3 colors if every color must be used at least once? problem2) prove that for any three sets A,B,C ; ((AB) U (BA))^ C = ((A^C) U (B^C)) (A^B^C) ^ means intersection Please explain the steps, help me understand. Thanks

Standard Combinatorics

Problem 1) We have 20 kinds of presents; and we have a large supply of each kind. We want to give presents to 12 children. It is not required that every child gets something; but no child can get 2 copies of the same present. In how many ways can we give presents? Problem 2) List all subsets of {a,b,c,d,e} containing {a.e} but not containing c Please for problem 2, dont just give me the answer, I can find it. What I cannot find though is a general answer. Here it seems more complicated. for {a.e} it doesn't seem to be just "from 5 objects choose 2" because we don't choose 2 random objects, but two particular objects. Please help understand. Thank you

[53 / (4-3)/4] - 12 + 1

[53 / (4-3)/4] - 12 + 1

5(3/3-4/4) -12 +1

5(3/3-4/4) -12 +1

5 ( 3/3 -4/4) -12 +1

5 ( 3/3 -4/4) -12 +1