Question

How many different photons (i.e., photons of different frequencies) can be emitted as a hydrogen atom in the third excited state falls back, directly or indirectly, to the ground state? Choose options 1 through 8.

Answer 1

The answer is 6.
The third excited state corresponds to n= 4 and the ground state is n= 1. So there are 3 transitions from the n= 4 state (n=4 to 3, n=4 to 2, n=4 to 1). The atoms that get to n=3 or n=2 can still keep emitting photons and there are 2 transitions from n=3 state (n=3 to 2, n=3 to 1). Finally, there is 1 transition from the n=2 state (n=2 to 1). In total there are 3+2+1=6 transitions.