Solution:

Given data:
- Energy level of the system: E0, 2E0, 3E0......
- Excited states are triply degenerate
- Four non-interacting bosons are placed in the system
- Total energy of these bosons: 5E0

We need to find the number of microstates.

To solve this problem, we need to apply the following steps:

Step 1: Calculate the maximum energy that can be occupied by the bosons

Since the total energy of the bosons is 5E0, the maximum energy that can be occupied by them is 3E0 (since the next energy level, 4E0, is greater than 5E0).

Step 2: Calculate the number of ways to distribute the energy among the bosons

We need to distribute the energy of 3E0 among the four bosons. To do this, we can use the following method:

- Start with the lowest energy level (E0) and distribute the energy among the bosons in all possible ways.
- Move to the next energy level (2E0) and repeat the process.
- Continue this process until we reach the maximum energy level that can be occupied (3E0).

For example, at the energy level E0, we have the following possibilities:

- All four bosons in the ground state: 1 way
- Three bosons in the ground state and one in the first excited state (2E0): 4 ways (since there are four bosons to choose from)
- Two bosons in the ground state and two in the first excited state: 6 ways
- One boson in the ground state and three in the first excited state: 4 ways
- All four bosons in the first excited state: 1 way

Similarly, we can calculate the number of possibilities at the other energy levels:

- At the energy level 2E0, we have the following possibilities: 1 way, 4 ways, 6 ways, 4 ways, 1 way
- At the energy level 3E0, we have the following possibilities: 1 way, 4 ways, 6 ways, 4 ways, 1 way

Step 3: Calculate the total number of microstates

To calculate the total number of microstates, we need to multiply the number of possibilities at each energy level:

Total number of microstates = (1 x 1 x 1) x (4 x 4 x 4 x 4 x 4 x 6 x 6 x 6 x 6 x 6 x 4 x 4 x 4 x 4 x 1 x 3 x 3 x 3 x 1) x (1 x 1 x 1) = 5

Therefore, the number of microstates is 5.