JEE Advanced 2025 Paper 1Question 1
Let the set of all relations $R$ on the set $\{a,b,c,d,e,f\}$, such that $R$ is reflexive and symmetric, and $R$ contains exactly 10 elements, be denoted by $\mathcal{S}$. Then the number of elements in $\mathcal{S}$ is \_\_\_\_\_\_\_\_.
Sets, Relations and Function
Answer$105$
Solution
Reflexivity uses all $6$ diagonal pairs; the remaining $4$ elements are $2$ symmetric pairs $\{(x,y),(y,x)\}$ chosen from $\binom62=15$, so the count is $\binom{15}{2}=105$.