A beam of electrons, each of energy 12.5 eV, is passed through hydrogen gas in its ground state, and the atoms excited by collisions are allowed to…

A colliding electron can give an atom any amount of energy up to its own, provided it matches a level. Excitation energies from $n = 1$: to $n=2$, $10.2\ \mathrm{eV}$; to $n=3$, $12.09\ \mathrm{eV}$; to $n=4$, $12.75\ \mathrm{eV}$. Since $12.5\ \mathrm{eV}$ lies between the last two, the highest level reached is $n = 3$. De-excitation from $n=3$ gives lines $3\to2$, $3\to1$, $2\to1$: $3$ lines. The shortest wavelength is for $3\to1$: $\lambda = \dfrac{1240}{12.09} \approx 103\ \mathrm{nm}$. Counting $n=4$ would give $6$ lines and $97\ \mathrm{nm}$, but that level is not accessible.