Introduction:
In this question, we are given a system consisting of N particles, where N is much greater than 1. Each particle can have one of two energies, E or E. We need to determine the average number of particles with energy E when the system is in equilibrium at temperature T.

Explanation:
To find the average number of particles with energy E, we can use the concept of equilibrium statistical mechanics. In equilibrium, the system is in a state where the number of particles with energy E is balanced by the number of particles with energy E.

1. Boltzmann Distribution:
The distribution of particles with different energies in equilibrium can be described by the Boltzmann distribution. According to the Boltzmann distribution, the probability of finding a particle with energy E is given by:

P(E) = (1/Z) * exp(-E/kT)

where P(E) is the probability, Z is the partition function, E is the energy of the particle, k is the Boltzmann constant, and T is the temperature.

2. Partition Function:
The partition function, Z, is a normalization constant that ensures the probabilities sum up to 1. It is defined as:

Z = Σ exp(-E/kT)

where the sum is taken over all possible energy states of the system.

3. Average Number of Particles:
The average number of particles with energy E can be obtained by multiplying the probability of finding a particle with energy E by the total number of particles, N. Therefore, the average number of particles with energy E is given by:

N(E) = N * P(E)

Substituting the Boltzmann distribution equation, we get:

N(E) = N * (1/Z) * exp(-E/kT)

Conclusion:
In conclusion, the average number of particles with energy E in the system at equilibrium and temperature T can be calculated using the Boltzmann distribution. This distribution provides the probability of finding a particle with a specific energy state. By multiplying this probability by the total number of particles in the system, we can obtain the average number of particles with energy E.