Combinations.

I had the following problem for discrete HW, and I can't think of a solution:

10 distinct pens are handed out to 4 distinct children, such that each child gets at least 2 pens. How many ways are there to do this?

10!/ (2!2!2!2!2!) * 4 * 4

The first part distributes the pens amongst the 4 children and leaves 2 pens left over handed to no-one. The second part gives the 9th and 10th pens to any of the four kids.


The longer way that simplifies to this answer is:

10C2 * 8C2 * 6C2 * 4C2 * 4 * 4

First 4 parts gives the pens to kids so that they each get two pens. 10 pens to begin with... choose 2 to gives to one kid. That leaves 8 pens... choose 2 to give to the second kid... and so on...

Now with two pens left... give one of those pens to any of the four kids. Then do the same with the last pen.