JEE Advanced · 2025 · 18 May 2025 · Paper 1 · Sets Relations and Functions

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 ____.

Answer: 105

Solution

A relation on a 6-element set has

6^{2} = 36

possible ordered pairs. For reflexivity, all 6 diagonal elements

(a, a), (b, b), \dots, (f, f)

must be included.

This leaves 30 off-diagonal elements, which form 15 symmetric pairs. For the relation to have exactly 10 elements total, we need

6 + 4 = 10

elements, meaning we select 2 symmetric pairs from the 15 available.

Number of ways:

(2 15) = 105

JEE Advanced · 2025 · 18 May 2025 · Paper 1 · Numerical answer