Let $\Omega, C, O$ be the set of total students(universal set), students who like their core branch and students who like other branches respectively.
$\underline{\text{Given:}}\ \ \ |\Omega| = 10000,\ \ | C^\complement \cap O^\complement | = 1500,\ \ |O| = 4|C|,\ \ |C \cap O| = 500$
$\underline{\text{To find:}}\ \ |C| = \ ?$
We know that,
$\begin{align*}
|C \cup O| &= |\Omega| - |(C \cup O)^\complement| \\
|C \cup O| &= |\Omega| - | C^\complement \cap O^\complement | && \text{(De Morgan's law)} \\
|C \cup O| &= 10000 - 1500 = 8500 \\
\end{align*}
$
Also,
$\begin{align*}
|C \cup O| &= |C| + |O| - |C \cap O| \\
|C| + |O| &= |C \cup O| + |C \cap O| \\
5|C| &= 8500 + 500 = 9000 \\
|C| &= 9000/5 = 1800
\end{align*}
$
So, the number of students who like their core branches is $1800$.
$\bf \therefore Ans = A.$