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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.

Solution Preview

Problem #1
We consider to give presents to 1 child. This child can get k kinds of present and each kind can have one copy.
Here k = 0, 1, 2, ..., 20. k = 0 means that this child gets nothing. For each k, we have C(20, k) ways to select presents.
C(20, k) = 20!/(k!(20-k)!) is the number of ways to select k items from 20 items. So totally a child has
C(20, 0) + C(20, 1) + ... + C(20, 19) ...

Solution Summary

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