Question

How many different ways are there to arrange 4 quanta among 3 atoms in a solid?

Short answer

The numbers of ways to arrange 4 quanta among 12 oscillations is 1,365.

Step-by-step solution

Step 1:

The number of oscillations among the 3 atoms is obtained below:

$N=4\times 3=12$

The numbers of oscillations among the 3 atoms is obtained by the multiply the 4 quanta among 12 oscillations.

Step 2:

The numbers of ways to arrange 4 quanta among 12 oscillations is obtained

below:

Let N represents the number of oscillations among the 3 atoms. That is, N = 12. Also ‘r’ represents the number of quanta. That is r = 4.

The numbers of ways is,

$$\begin{gathered} n=\frac{\left(N+r+1\right)!}{r!\left(N+1\right)!} \\ =\frac{\left(12+4-1\right)!}{\left(12-1\right)!\left(4\right)!} \\ =1,365 \end{gathered}$$