And E2, where E1 < e2.="" the="" total="" energy="" of="" the="" system="" is="" given="" by="" the="" sum="" of="" the="" energies="" of="" all="" the="" particles.="" let="" n1="" and="" n2="" be="" the="" number="" of="" particles="" in="" energy="" levels="" e1="" and="" e2,="" />

In an isolated system, the total energy is conserved, which means that the total energy before and after any process remains the same. Therefore, the total energy of the system can be expressed as:

Total energy = N1 * E1 + N2 * E2

Since each particle can occupy only one energy level, the total number of particles N is equal to the sum of N1 and N2:

N = N1 + N2

Now, let's consider the number of ways to distribute N distinguishable particles into N1 and N2 energy levels. This can be calculated using combinations.

The number of ways to choose N1 particles out of N is given by:

Number of ways = N! / (N1! * (N - N1)!)

Similarly, the number of ways to choose N2 particles out of N is given by:

Number of ways = N! / (N2! * (N - N2)!)

Therefore, the total number of ways to distribute N particles into N1 and N2 energy levels is given by the product of the two combinations:

Total number of ways = (N! / (N1! * (N - N1)!)) * (N! / (N2! * (N - N2)!))

Finally, the probability of a particular distribution of particles with N1 and N2 can be calculated as the ratio of the number of ways to distribute N particles into N1 and N2 energy levels to the total number of ways to distribute N particles:

Probability = (Number of ways to distribute N particles into N1 and N2 energy levels) / (Total number of ways to distribute N particles)

Probability = [(N! / (N1! * (N - N1)!)) * (N! / (N2! * (N - N2)!))] / [(N! / (N1! * (N - N1)!)) * (N! / (N2! * (N - N2)!))]

Probability = 1

This means that any particular distribution of particles into N1 and N2 energy levels is equally probable in an isolated system.