What is the combinatoric describing the number of ways to place $N$ items in $K$ bins where each bin has at least $1$ item? Is it just $N-1$ choose $K-1$?
We know the number of nonnegative solutions to $x_1+x_2+\cdots+x_K=N$ is
$(N+K-1)!/N!(K-1)!$ If you want $x_i \geq 1$ for all i. Then we can set $x_i=y_i+1$, with $y_i \geq 0$
and count the number of nonnegative solutions to $(y_1+1)+(y_2+1)+\cdots+(y_K+1)=N$; that is,
$y_1+y_2+\cdots+y_K=N-K$. Thus, we get $(N-1)!/(K-1)!(N-K)!$ which is indeed N-1 choose K-1.