Introduction:
In this system, we have five weekly interacting distinguishable particles that can occupy three energy levels: 0E, E, and 2E. The total energy of the system is 5E. We need to determine all possible macrostates and their corresponding number of microstates, and calculate the entropy of the system.

Macrostates and Microstates:
A macrostate refers to the overall configuration of the system, while a microstate represents a specific arrangement of particles in energy levels. In this case, we have five particles and three energy levels.

Let's consider all possible macrostates and calculate the number of microstates for each:

1. Macrostate: {0E, 0E, 0E, 0E, 5E}
In this macrostate, all particles are in the lowest energy level (0E). The number of microstates can be calculated using the formula for combinations: C(n+r-1, r) where n is the number of particles and r is the number of energy levels.
Number of microstates = C(5+3-1, 3) = C(7, 3) = 35

2. Macrostate: {0E, 0E, 0E, E, 4E}
In this macrostate, four particles are in the lowest energy level (0E) and one particle is in the second energy level (E).
Number of microstates = C(5+2-1, 2) = C(6, 2) = 15

3. Macrostate: {0E, 0E, 0E, 2E, 3E}
In this macrostate, three particles are in the lowest energy level (0E), one particle is in the second energy level (2E), and one particle is in the third energy level (3E).
Number of microstates = C(5+3-1, 3) = C(7, 3) = 35

4. Macrostate: {0E, 0E, E, E, 3E}
In this macrostate, two particles are in the lowest energy level (0E), two particles are in the second energy level (E), and one particle is in the third energy level (3E).
Number of microstates = C(5+2-1, 2) = C(6, 2) = 15

5. Macrostate: {0E, E, E, E, 2E}
In this macrostate, one particle is in the lowest energy level (0E), three particles are in the second energy level (E), and one particle is in the third energy level (2E).
Number of microstates = C(5+2-1, 2) = C(6, 2) = 15

6. Macrostate: {E, E, E, E, E}
In this macrostate, all particles are in the second energy level (E).
Number of microstates = C(5+1-1, 1) = C(5, 1) = 5

7. Macrostate: {0E, 2E, 2E, 2E, 0E}
In this macrostate