Past exam of the laser science and quantum information processing course of the University of Oxford / 2019 / 1 / c / iv / Solution

Extracting 95 percent of the maximum available inversion energy means $\eta =0.95/\beta$, or $x=\beta \eta =0.95$. Substitution in ii givesThe degeneracy factor cancels because the target is a fraction of the maximum energy $N_i^{\ast }{V}_g\hslash {\omega }_L/\beta$, rather than a fraction of $N_i^{\ast }{V}_g\hslash {\omega }_L$ without that factor.