JEE Advanced 2025 — Paper 1 — MATH-III solution

JEE Advanced2025Paper 1MATH-III

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 $S$ .

Then the number of elements in $S$ is __________.

Your answer

Solution

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 $(2 6) = 15$ 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 $(2 15) = 105.$

$∴$ The answer is $105$ .

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