Question

Verify that this equation gives the correct number of ways to arrange 0, 1, 2, 3, or 4 quanta among 3 one-dimensional oscillators, given in earlier tables (1, 3, 6, 10, 15).

Q1	Q2	#ways1	#ways2	#ways1 #ways2
0	4	1	15	15
1	3	3	10	30
2	2	6	6	36
3	1	10	3	30
4	0	15	1	15

Short answer

The equation gives the correct number of ways to arrange4quanta among 3 one-dimensional oscillators.

Step-by-step solution

Identification of given data

• The given number to arrange is 0,1,2,3,4 quanta.

Concept of permutation

A sequential or linear arrangement of its members, or if the set is already sorted, a reordering of its pieces. An act or procedure of altering the overall sequential order of a predetermined set is referred to as a "permutation."

Determination of the number of ways to arrange quanta in one-dimensional oscillators

The number of ways microstates can be evaluated using

$\mathrm{\Omega }=\frac{\left(\mathrm{q}+\mathrm{N}-1\right)!}{\mathrm{q}!\left(\mathrm{N}-1\right)!}$

Substitute values in the above equation,

$$\begin{gathered} \mathrm{\Omega }=\frac{\left(4+3-1\right)!}{4!\left(3-1\right)!} \\ \Omega =\frac{6!}{4!2!} \\ \Omega =15 \end{gathered}$$

Thus, the result is the same in the equation and tabular form.