TITLE: How many ways are there to give cookies and candies to these kids?
QUESTION [0 upvotes]: The same kind of 5 cookies are given to 3 kids. At least one cookie is given to each
child. Also, the same kind of 5 candies are given to the kids who received only one
cookie. How many ways are there to give cookies and candies to these children?
My Work: C(5,3) x C(5,3) = 10 x 10 = 100 ways
C(5,3) (One for the cookies)
C(5,3) (One for the candies)
Did I do this correctly? I am not sure and want to double check if my reasoning is correct.

REPLY [1 votes]: There are two different partitions for the cookies: $2, 2, 1$ and $3, 1, 1$.
The partition $2, 2, 1$ can be distributed in three ways (each child can the one with 1 cookie).  The candies must all be given to this child, so we have a total of three divisions.
The partition $3, 1, 1$ can be distributed in three ways (each child can be the one with 3 cookies).  For each of these three, there are four ways of distributing the candies to the other two children: each of the two children can be the child to receive more, and the split can be either $4,1$ or $3,2$.  As such, there are a total of twelve divisions for this cookie partition.
In total, there are 15 ways we can divide the cookies and candies
{"set_name": "stack_exchange", "score": 0, "question_id": 3775084}