Combination with repetition and no repetition

Question

If a father has $n$ children and will generously give an expensive gift to each child.

1) Assume that the number of children is a multiple of $5$. If there are $n/5$ kinds of gifts and $5$ of each type, how many ways are there of giving out gifts?

2) If the children all look alike to their father, and there are $5$ kinds of gifts, unlimited supply of each kind, how many ways are there of giving out gifts (from the father's point of view)?

For 1)Since there are $5$ of each type of $n/5$ kinds of gifts, I was thinking it would be $5(n/5!)^{5n}$ for the 1st child and $4(n/5!)^{5n}$ for the second child. For 2) since there are repetition for the gifts I have $(n/5!)^5$ –

Answer 1

Hints:

1) Think about the Multinomial coefficient.

2) The first child will receive one of the 5 possible types of gift. And the other children, how many possibilities each one have?

Answer 2

1) there are n gifts total to be distributed among n. So actually you have to partition n children into n/5 groups. This is done in $\frac {n!} {(n/5)!^5} $ ways. Now distribute 5 kinds of gifts in these 5 groups. So multiply by 5! 2) let $x_i$ denote the no. Of gifts of ith kind distributed among children. (here all children are identical) So you have $x_1+x_2+x_3+x_4+x_5= n$ where $x_i \ge 0 $and each is an integer. The number of solutions of this integral equation which is $C_4^{n+4}$