define F(n, k) as the way to put n mathematicians in k rooms with at least 1 M in a room (so for n < k, F(n, k) = 0)
F(n, 1) = 1; F(k, k) = k!;
F(n, k) = C(n, 1)*F(n-1, k-1) + C(n, 2)*F(n-2, k-1) + .. C(n, i)*F(n-i, k-1) + ... + C(n, n - k + 1 )*F(k-1, k-1); C(n, i)*F(n-i, k-1) means choose i...