Relations
PYP_JEE_ADV_2025_P1
Grade None
Question:
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 ___.
Step-by-Step Solution
Key Concept: Reflexive symmetric relations: diagonal mandatory + symmetric off-diagonal pairs
A reflexive symmetric relation on a set of $n=6$ elements must include all 6 diagonal pairs $(a,a),(b,b),\ldots,(f,f)$. The remaining elements are chosen from the $\binom{6}{2}=15$ off-diagonal symmetric pairs $\{(x,y),(y,x)\}$ (each such pair contributes 2 elements).
We need exactly 10 elements total. The 6 reflexive pairs are mandatory, so we need $10 - 6 = 4$ more elements from off-diagonal pairs. Since off-diagonal pairs come in symmetric pairs of 2, we need exactly $4/2 = 2$ off-diagonal symmetric pairs.
Number of ways to choose 2 pairs from 15: $\binom{15}{2} = 105$.
Correct Answer: 105