Pauli's Exclusion Principle states that no two electrons in an atom can have the same four quantum numbers $(n,l,m,s)$.

• Given: $n=4$ and $m=-2$.
• For $n=4$, the allowed values of $l$ are $0,1,2,3$.
• For $m=-2$, the allowed values of $l$ must satisfy $|m|\le l$, so $l$ can be $2$ ($4d$) or $3$ ($4f$).
• This gives two distinct orbitals with $(n=4,m=-2)$: the $(4,2,-2)$ orbital in the $4d$ subshell and the $(4,3,-2)$ orbital in the $4f$ subshell.
• Each orbital can accommodate a maximum of 2 electrons (with opposite spins, $s=+1/2$ or $-1/2$).
• Therefore, the maximum number of electrons with $n=4$ and $m=-2$ is $2+2=4$.