The number of orbitals associated with quantum numbers $n=5$, $m_s=+\frac{1}{2}$  is ____

To answer the number of electrons with quantum numbers n=5 and ms=+1/2, you would need to consider all possible l values (l=0,1,2,3,4), and all possible ml values for each l (ml= -l...0...l). The fast way to do it is also to use the relation that the number of orbitals within the n shell of electrons is n². So for principle quantum number n=5, there are 25 orbitals, and only 1 electron within each orbital can have a value m_s= +1/2, meaning there are 25 electrons with n=5 and m_s= +1/2.