This is probably a simple thing to do but it is driving me up the wall.(adsbygoogle = window.adsbygoogle || []).push({});

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 = 2^{3}.

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 = 2^{2}.

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 = 2^{4}.

From this I can see that P_t = 2^{M}, 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A quick question about multiplicity?

**Physics Forums | Science Articles, Homework Help, Discussion**