Thread: Configuration of particles

Suppose that n distinguishable particles are placed randomly in N boxes (states). A particular configuration of this system is such that there are ns particles in state s, where 1<=s<=N. If the ordering of particles in any particular state doesn't matter, find the number of ways of realising a particular configuration.

It's not at all obvious to me what I should do to tackle this question, any help would be hugely appreciated.

Originally Posted by free_to_fly

Suppose that n distinguishable particles are placed randomly in N boxes (states). A particular configuration of this system is such that there are ns particles in state s, where 1<=s<=N. If the ordering of particles in any particular state doesn't matter, find the number of ways of realising a particular configuration.

It's not at all obvious to me what I should do to tackle this question, any help would be hugely appreciated.

Hi Free To Fly,

If I understand your question correctly, the number of ways is the multinomial coefficient,

.

Thanks for the answer, but could you please explain where you got it from? I can't really see that. Thanks.

The multinomial coefficient



counts the number of ways to put n distinct objects in N boxes, with in the first box, in the second box, etc. See Multinomial theorem - Wikipedia, the free encyclopedia.

For a derivation, consider all permutations of the n objects and consider the first to be in the first box, the next to be in the second box, etc. The order of the objects in the boxes is considered irrelevant, however, so divide by to compensate for over-counting in box 1, divide by to compensate for over-counting in box 2, ...

It's just like the binomial coefficient, only generalized to N choices instead of 2.