I'm doing some homework and one of the questions is is as follows. For m $\ge$ 3 show that S(m, m - 2) = $\frac{1}{24}$ m(m-1)(m-2)(3m-1).
My interpretation of the question is as follows. This problem can be modelled as the number of ways to distribute m objects amongst m - 2 containers with none left empty. This can be split into two cases:
- Three objects in one container, with the other m - 3 containers each containing one.
- Two containers each contain two objects, with the other m - 4 containing one.
Case one can happen in $\dbinom{m}{3}$ ways (choose three objects to put into one container). For case two I got $\dbinom{m - 2}{2}\dbinom{m}{2}$ ways to distribute them.
I then go through the algebra but don't get the expected result. This leads me to believe the counting I did above is wrong.
What is the correct way to model the two cases? Is what I did wrong or did I just make a mistake with my algebra? Thanks