How many spectral lines are emitted by atomic hydrogen excited to the `n-th` energy level?

Explanation

Because of cascading all possible transition are seen. Thus we look for the number of was in which we can select upper and lower levels. The number of ways we can do this is
`(1)/(2_n(n-1)`
where the factor `(1)/(2)` takes account of the fact that the photon emission always arise from upper `rarr` lower transition.