Find the number of ways of putting$\mathbf{3}$particles in $5$boxes according to the three kinds of statistics.

The three particles are in five boxes.

If a given experiment's sample space is known to be uniform, the probability of an event can be calculated using the event sizes and the sample space.

Use the Maxwell Boltzmann method. In this method, particles are distinguishable so the number of methods arranged is given below.

$\begin{array}{c}{M}_1=n^N\\ =5^3\\ =125\end{array}$

Use the Fermi-dirac method. In this method, particles are not distinguishable so the number of methods arranged is given below.

$\begin{array}{c}{M}_2=C(n,N)\\ =C(5,3)\\ =10\end{array}$

Use the Bose Einstein method. In this method, particles are not distinguishable but allow to put 2 balls in the same box. The number of methods arranged is given below.

$\begin{array}{c}{M}_3=C(n-1+N,N)\\ =C(7,3)\\ =35\end{array}$

Hence, the required values are given below.

$\begin{array}{c}{M}_1=125\\ M_2=10\\ M_3=35\end{array}$