Suppose n_p and n_e denote the numbers of holes and conduction...

In an intrinsic semiconductor, the number of holes is equal to the number of conduction electrons, giving $n_p = n_e$.

An extrinsic semiconductor is formed by doping an intrinsic semiconductor with either pentavalent or trivalent impurities.

For an n-type semiconductor, electrons are the majority charge carriers, so $n_e > n_p$.

For a p-type semiconductor, holes are the majority charge carriers, so $n_p > n_e$.

Therefore, in any extrinsic semiconductor, the number of holes is never equal to the number of conduction electrons, which implies $n_p \neq n_e$.