I recently received the following interesting problem from Shadowcat, which is a generalization of the cookie problem I’ve written about previously. We again want to count the ways to distribute identical cookies to non-identical students, with the twist that we impose an upper bound on the number of cookies received by each student (quite reasonable if we want to be mindful of the students’ nutrition):
Imagine that instead of ten cookies and five students, you have fifty cookies and ten students. (It’s easier to quantify this situation using larger numbers.) How many ways can you distribute these cookies among the students so that no student has any more than ten cookies?
Students may be given any number of cookies less than or equal to ten, including zero. The cookies are identical, just as in the original problem, so, as with the original problem, it doesn’t matter which cookie the student gets, just how many. But the students, again, are *not* identical, so which student gets a specific number of cookies is important.
I unfortunately haven’t had much time to think about it yet. Feel free to leave comments, thoughts, partial solutions, and solutions in the comments!