Here’s a neat problem I saw in a recent post by Steven Miller on the Williams College math department blog. The problem comes from an old Putnam competition, one of the most prestigious college mathematics competitions. (It’s also one of the most difficult—out of a possible 120 points, the median score is 1!)
There are ten identical cookies and five students. How many ways can the cookies be distributed among the students?
Note that the cookies are identical, so it doesn’t matter which cookies a student gets, just how many. The students, of course, are not identical, so student A getting four cookies and student B getting two is different than A getting two and B getting four. Assume that the cookies can’t be split into pieces. Note that giving cookies to some students but not others is a valid way of distributing them, as long as all the cookies are distributed. For example, giving six cookies to student A, four to student C, and none to anyone else is a valid distribution.
My wife and I had a fun time solving this problem, which leads to all kinds of interesting combinatorial insights. I’ll describe our
analysis in an upcoming post.