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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
E_{1}=0, E_{2}=E, E_{3}=4E
Let us first fill the E_{1} 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 E_{1} is
N(N-1)(N-2)=\frac{N!}{(N-3)!}
Generally for n_{1} particles placed in E_{1} is,
\frac{N!}{n_{1}!(N-1)!}
for E_{2} state,
\frac{(N-n_{1})!}{n_{2}!(N-n_{1}n_{2})!}
for E_{3} state,
\frac{(N-n_{1}n_{2})!}{n_{3}!(N-n_{1}n_{2}n_{3})!}
Total number of particles in all three state will be
P=\frac{N!}{n_{1}!n_{2}!n_{3}!}
Substituting values
P=\frac{N!}{0!1!4!}
Am I on right track?