Permutation and combination. In how many ways can 12 indistinguishable apples and 1 orange be distributed among 3 distinguishable children in such a way that each child gets at least one fruit?
What if the apples are different (distinguishable)?

HINT: For the first problem, use a stars-and-bars calculation to find the number of ways to distribute the $12$ apples so that each child gets at least one, then multiply by $12$ to account for the fact that the orange can be given to any child. Then count the ways in which one child gets just the orange, and the apples are distributed to the other two children in such a way that each of the two gets at least one apple. You want the sum of these two numbers.
For the second problem you want a Stirling number of the second kind.
