Let $A$ be the event in which the selected student has opted for $NCC$ and $B$ be the event in which the selected student has opted for $NSS$.

Total number of students $=60$

Number of students who have opted for $NCC =30$

$\therefore $ $P(A)=\frac{30}{60}=\frac{1}{2}$

Number of students who have opted for $NSS =32$

$\therefore $ $P(B)=\frac{32}{60}=\frac{8}{15}$

Number of students who have opted for both $NCC$ and $NSS = 24$

$\therefore $ $P ( A$ and $B )=\frac{24}{60}=\frac{2}{5}$

We know that $P ( A $ or $ B )= P ( A )+ P ( B )- P ( A$ and $B )$

$\therefore $ $P ( A$ or $B )=\frac{1}{2}+\frac{8}{15}-\frac{2}{5}=\frac{15+16-12}{30}=\frac{19}{30}$

Thus, the probability that the selected student has opted for $NCC$ or $NSS$ is $\frac{19}{30}$