JEE Advanced 2025 Paper 1 · Q08 · Permutations & Combinations

Question

Let $S$ be 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. Find $|S|$.

SolveKar AI · Accepted Answer

A reflexive relation must contain all $6$ diagonal pairs $(a,a), (b,b), \ldots, (f,f)$. So $6$ elements are forced; we need exactly $10$ total, hence $4$ more elements off-diagonal. Symmetry forces off-diagonal pairs to come in pairs $(i,j)$ and $(j,i)$. A single 'off-diagonal pair' $(i,j)$ with $i 
e j$ forces the symmetric partner $(j,i)$, contributing $2$ elements. To add exactly $4$ more elements, we need exactly $2$ such unordered pairs $\{i,j\}$ from the $6$-element set. Number of unordered pairs: $\binom{6}{2} = 15$. Choose $2$ of them: $\binom{15}{2} = 105$. Answer: $105$.

JEE Advanced 2025 Paper 1 · Q08 · Permutations & Combinations

Solution