A hydrogen atom in its ground state absorbs a photon of energy $12.75\,\text{eV}$. The maximum number of distinct spectral lines that can subsequently be emitted is

Find the highest level the photon can reach: $E_n - E_1 = 12.75\,\text{eV}$, so $E_n = -13.6 + 12.75 = -0.85\,\text{eV}$. From $-13.6/n^2 = -0.85$, $n^2 = 16$, $n = 4$. From $n=4$, transitions down can land on $n=3,2,1$ — total distinct lines $= \binom{4}{2} = 6$.