A relation consists of set of ordered pairs $(a,b)$. Here $a$ can be chosen in $n$ ways and similarly $b$ can be chosen in $n$ ways. So, totally $n^2$ possible ordered pairs are possible for a relation. Now each of these ordered pair can either be present in the relation or not- 2 possibilities for each of the $n^2$ pair. So, total number of possible relations = $$2^{\left(n^2\right)}$$
Now, for a relation $R$ to be reflexive, ordered pairs $\left\{(a,a) \mid a \in S \right\}$, must be present in $R$. i.e.; the relation set $R$ must have $n$ ordered pairs fixed. So, number of ordered pairs possible is $ n^2 - n$ and hence total number of reflexive relations is equal to $$ 2^{\left(n^2-n\right)}$$
for $n=5$, answer will be, $2^{5^2-5}=2^{20}$
Therefore, option C is correct