Answer
There are $7$ different frequencies of light.
Work Step by Step
The quantized energies for an electron trapped in a three-dimensional infinite potential well that forms a e rectangular box are
$E_{nx,ny,nz}=\frac{h^2}{8m}\Big(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}\Big)$,
where m is the electron mass and $n_x$ is a quantum number for well width$ L_x$, $n_y$ is a quantum number for well width $L_y$ and $n_z$ is a quantum number for well width $L_z$.
In this problem, the electron is contained in a cubical box of widths $L_x=L_y=L_z=L$.
$\therefore\;\;E_{nx,ny,nz}=\frac{h^2}{8mL^2}\Big(n_x^2+n_y^2+n_z^2\Big)$
The sates that correspond to the lowest five energy levels are
$1.$ Ground sate: $(1,1,1)$
$2.$ First excited state with 3 fold degeneracy: $(2,1,1)$, $(1,2,1)$ and $(1,1,2)$
$3.$ Second excited state with 3 fold degeneracy: $(2,2,1)$, $(1,2,2)$ and $(2,1,2)$
$4.$ Third excited state with 3 fold degeneracy: $(3,1,1)$, $(1,3,1)$ and $(1,1,3)$
$5.$ Fourth excited state: $(2,2,2)$
Thus the lowest five energy levels are
$E_{1,1,1}=3\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,1,1}=E_{1,2,1}=E_{1,1,2}=6\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,2,1}=E_{1,2,2}=E_{2,1,2}=9\Big(\frac{h^2}{8mL^2}\Big)$
$E_{3,1,1}=E_{1,3,1}=E_{1,1,3}=11\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,2,2}=12\Big(\frac{h^2}{8mL^2}\Big)$
Now the possible transition with the corresponding energy difference are given below:
$\text{Transition}\;\;\;\;\;\;\;\;\text{Energy difference}(\Delta E)$
$E_{2,2,2}\leftrightarrow E_{1,1,1}\;\;\;\ \;\;\;\;\;\;\;\; 9\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,2,2}\leftrightarrow E_{2,1,1}\;\;\;\ \;\;\;\;\;\;\;\; 6\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,2,2}\leftrightarrow E_{2,2,1}\;\;\;\ \;\;\;\;\;\;\;\; 3\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,2,2}\leftrightarrow E_{3,1,1}\;\;\;\ \;\;\;\;\;\;\;\; 1\Big(\frac{h^2}{8mL^2}\Big)$
$E_{3,1,1}\leftrightarrow E_{1,1,1}\;\;\;\ \;\;\;\;\;\;\;\; 8\Big(\frac{h^2}{8mL^2}\Big)$
$E_{3,1,1}\leftrightarrow E_{2,1,1}\;\;\;\ \;\;\;\;\;\;\;\; 5\Big(\frac{h^2}{8mL^2}\Big)$
$E_{3,1,1}\leftrightarrow E_{2,2,1}\;\;\;\ \;\;\;\;\;\;\;\; 2\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,2,1}\leftrightarrow E_{1,1,1}\;\;\;\ \;\;\;\;\;\;\;\; 6\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,2,1}\leftrightarrow E_{2,1,1}\;\;\;\ \;\;\;\;\;\;\;\; 3\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,1,1}\leftrightarrow E_{1,1,1}\;\;\;\ \;\;\;\;\;\;\;\; 3\Big(\frac{h^2}{8mL^2}\Big)$
Form the energy difference value, we can say that there are $7$ different frequencies of light.
