This is probably a simple thing to do but it is driving me up the wall. Say I had a box with M slots, and there are N particles inside this box, and each slot can hold, at most, 1 particle. (where N is less than or equal to M). I am trying to calculate the multiplicity of this box system by counting the number of possibilities of distributing my N particles over these M slots. I have tried to make a simple example, with M = 3. If I first take N=0, then there is only 1 configuration, ie. the multiplicity (lets say, P) is 1. I have drawn out little diagrams and determined that: P(N=0) = 1 P(N=1) = 3 P(N=2) = 3 P(N=4) = 1 The total sum of these multiplicities P_t = 1+3+3+1 = 8 = 23. I have done the same thing for M = 2: P(N=0) = 1 P(N=1) = 2 P(N=2) = 1 Here P_t = 1+2+1 = 4 = 22. and M = 4: P(N=0) = 1 P(N=1) = 4 P(N=2) = 6 P(N=3) = 4 P(N=4) = 1 Here, P_t = 1+4+6+4+1 = 16 = 24. From this I can see that P_t = 2M, and that if N=0 or N=M, then P=1. However, I need a general expression, since I do not want to draw out these little diagrams for higher and higher M or N. In other words, how do I find out an expression for the multiplicity P=P(N,M)? I cannot spot a pattern just from the examples I have done. Out of M slots, how many ways are there of arranging the N particles I happen to have inside the box (N</= M) ? Thanks.