$(a)$ How many sub-shells are associated with $n=4?$
$(b)$ How many electrons will be present in the sub-shells having $m_{s}$ value of $-1/2$ for $n=4?$

(N/A) For a given principal quantum number $n$,the possible values of the azimuthal quantum number $l$ range from $0$ to $(n-1)$.
For $n=4$,$l = 0, 1, 2, 3$.
These correspond to the $4s, 4p, 4d$,and $4f$ sub-shells. Thus,there are $4$ sub-shells.
$(b)$ The total number of orbitals in a shell with principal quantum number $n$ is given by $n^2$.
For $n=4$,the total number of orbitals $= 4^2 = 16$.
According to the Pauli exclusion principle,each orbital can hold a maximum of $2$ electrons with opposite spins ($m_s = +1/2$ and $m_s = -1/2$).
Therefore,in $16$ orbitals,there will be $16$ electrons with $m_s = -1/2$.