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## Homework Statement

A system of N particles has three possible energy levels namely; 0, E and 4E. How many particles does one expect in the second state at temperature T?

## Homework Equations

It's a sample problem for our finals. Our Text book is

*Statistical Mechanics by Roger Bowley and Mariana Sanchez*.

## The Attempt at a Solution

Three Energy levels

[itex]E_{1}=0[/itex], [itex]E_{2}=E[/itex], [itex]E_{3}=4E[/itex]

Let us first fill the [itex]E_{1}[/itex] state with 3 particle.

N distinguishable ways of selecting the first particle

N-1 different ways to select second particle

N-2 different ways to select third particle

So the total number of ways to place first three particles in state [itex]E_{1}[/itex] is

[itex]N(N-1)(N-2)=\frac{N!}{(N-3)!}[/itex]

Generally for [itex]n_{1}[/itex] particles placed in [itex]E_{1}[/itex] is,

[itex]\frac{N!}{n_{1}!(N-1)!}[/itex]

for [itex]E_{2}[/itex] state,

[itex]\frac{(N-n_{1})!}{n_{2}!(N-n_{1}n_{2})!}[/itex]

for [itex]E_{3}[/itex] state,

[itex]\frac{(N-n_{1}n_{2})!}{n_{3}!(N-n_{1}n_{2}n_{3})!}[/itex]

Total number of particles in all three state will be

[itex]P=\frac{N!}{n_{1}!n_{2}!n_{3}!}[/itex]

Substituting values

[itex]P=\frac{N!}{0!1!4!}[/itex]

Am I on right track?