JEE Advanced 2025 Paper 1, Mathematics Section 3 Q1: Relations

On a set with $6$ elements, reflexivity forces all $6$ diagonal pairs $(x,x)$ to lie in $R$. Off the diagonal there are $6\times 5=30$ ordered pairs, which group into $\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$ unordered pairs $\{x,y\}$. Symmetry means each chosen off-diagonal pair contributes exactly $2$ ordered elements to $R$.

To have $|R|=10$, we need $10-6=4$ additional ordered pairs, which is exactly $2$ unordered pairs from the $15$ available.

The number of choices is $\left(\genfrac{}{}{0ex}{}{15}{2}\right)=105.$

$\therefore$ The answer is $\boxed{105}$.