Consider a system of 5 non-interacting distinguishable particles. Each...
Let's label the energy levels as 1 and 2. Since there are 5 particles, the total number of possible configurations is 2^5 = 32. We can list all the possible configurations as follows:
1. All particles in level 1: 11111
2. One particle in level 2, rest in level 1: 11112, 11121, 11211, 12111, 21111
3. Two particles in level 2, rest in level 1: 11122, 1112, 11212, 12112, 21112, 1122, 1212, 2112, 22211, 22112, 21212
4. Three particles in level 2, rest in level 1: 111222, 11122, 11222, 12222, 22211, 22112, 21212, 1222, 2122, 2212
5. Four particles in level 2, rest in level 1: 1112222, 111222, 112222, 122222, 22221, 22212, 22122, 21222
6. All particles in level 2: 22222
Note that we can count the number of configurations in each category by using the binomial coefficient formula:
n choose k = n!/(k!(n-k)!)
where n is the total number of particles and k is the number of particles in level 2. For example, the number of configurations with one particle in level 2 is:
5 choose 1 = 5!/(1!(5-1)!) = 5
Similarly, the number of configurations with two particles in level 2 is:
5 choose 2 = 5!/(2!(5-2)!) = 10
We can use this formula to count the number of configurations in each category and add them up to get the total number of configurations.