A beam of electrons, each of kinetic energy 12.5 eV, is passed through a sample of hydrogen atoms in the ground state. The atoms that get excited…

An electron of $12.5\ \mathrm{eV}$ can excite a ground-state atom only to a level with $13.6(1-1/n^2)\le 12.5$. For $n=3$ the requirement is $12.09\ \mathrm{eV}$, which is allowed, but for $n=4$ it is $12.75\ \mathrm{eV}$, which exceeds $12.5\ \mathrm{eV}$. So the highest level reached is $n=3$, and the possible lines are $3\to1$ ($12.09\ \mathrm{eV}$), $2\to1$ ($10.20\ \mathrm{eV}$) and $3\to2$ ($1.89\ \mathrm{eV}$). The photoelectric condition is $h\nu>\phi=2.0\ \mathrm{eV}$. The first two lines satisfy it, whereas $1.89\ \mathrm{eV}<2.0\ \mathrm{eV}$ fails. Hence exactly $2$ lines eject photoelectrons. The trap is to use $n=4$ and count six lines, or to forget that the $3\to2$ photon is too weak.