Final answer:
For system A, there are 3 energetically equivalent ways to distribute the particles. For system B, there are 12 energetically equivalent ways to distribute the particles.
Step-by-step explanation:
To find the number of energetically equivalent ways to distribute the particles in system A, we need to consider the different arrangements of the particles that result in the same total energy. In System A, the particles have energies of 10 J, 10 J, and 10 J. We can rearrange these energies in different ways. To determine the number of arrangements, we can use the concept of permutations. The number of arrangements can be calculated using the formula:
n! / (n₁! * n₂! * n₃!)
where n is the total number of particles and n₁, n₂, n₃ are the number of particles with each energy. In system A, we have three particles, each with 10 J of energy, so n = 3, n₁ = 1, n₂ = 1, n₃ = 1. Plugging these values into the formula, we get:
3! / (1! * 1! * 1!) = 3
Therefore, there are 3 energetically equivalent ways to distribute the particles in system A.
For system B, we have four particles with energies of 12 J, 10 J, 10 J, and 8 J. Using the same formula, we can calculate the number of arrangements:
4! / (1! * 2! * 1!) = 12
Therefore, there are 12 energetically equivalent ways to distribute the particles in system B.