In how many ways Rdistinct balls can be placed in M distinct cells?

Question

R is greater than M and all cells must contain balls.

Answer 1

Solve the problem:

$$x_1+x_2+...+x_M=R$$

with $x_i \ge 1$ and then you can do $x_i=y_i+1$, and the equation became:

$$y_1+y_2+...+y_M=R-M$$

with $y_i \ge 0$.

If the balls are different we will have:

$$\frac{(R-M+M-1)!}{(M-1)!}=\frac{(R-1)!}{(M-1)!}$$