The possible number of microstates refers to the number of different ways in which the particles can be distributed among the available energy states. In this case, we have three spinless fermions and four available energy states.



Fermions are a class of particles that follow the Pauli Exclusion Principle. This principle states that no two identical fermions can occupy the same quantum state simultaneously. As a result, each fermion must occupy a unique energy state.



To determine the possible number of microstates, we can consider the distribution of the three fermions among the four available energy states.

Case 1: All Fermions in Different Energy States
In this case, each fermion occupies a unique energy state. There are four possible ways to distribute the first fermion, three possible ways for the second fermion, and two possible ways for the third fermion. Therefore, the number of microstates for this case is 4 x 3 x 2 = 24.

Case 2: Two Fermions in the Same Energy State
In this case, two fermions occupy the same energy state, while the third fermion occupies a different energy state. There are four possible ways to choose the energy state for the third fermion, and then two possible ways to distribute the remaining two fermions among the other three energy states. Therefore, the number of microstates for this case is 4 x 2 x 3 = 24.

Case 3: All Fermions in the Same Energy State
In this case, all three fermions occupy the same energy state. There are four possible energy states to choose from. Therefore, the number of microstates for this case is 4.



To find the total number of microstates, we sum up the number of microstates for each case:

Total number of microstates = 24 + 24 + 4 = 52

Therefore, the possible number of microstates when three spinless fermions are distributed in four available energy states is 52.