Let \(\mathrm{S}=\{1,2,3, \ldots, 10\}\). Consider the set \(\mathrm{X}=\{\mathrm{R}: \mathrm{R}\) is an equivalence relation on the set S such that R has exactly 42 elements \(\}\). Then the number of elements in X is ____.

JEE Advanced · 2026 · 17 May 2026 · Paper 1 · Relations And Functions XII

Let

S = {1, 2, 3, \dots, 10}

. Consider the set

X = {R : R

is an equivalence relation on the set S such that R has exactly 42 elements

}

.
Then the number of elements in X is ____.

Answer: 2520

An equivalence relation partitions the set into equivalence classes. If the partition has classes of sizes

n_{1}, n_{2}, \dots, n_{k}

, then the number of elements in the relation is

\sum n_{i}^{2}

.

We need partitions of 10 where

\sum n_{i}^{2} = 42

.

Testing:

6^{2} + 2^{2} + 1^{2} + 1^{2} = 36 + 4 + 1 + 1 = 42

✓

5^{2} + 4^{2} + 1^{2} = 25 + 16 + 1 = 42

✓

For partition (6,2,1,1): Number of ways =

\frac{10 !}{6 ! 2 ! 1 ! 1 !} \cdot \frac{1}{2 !} = 1260

For partition (5,4,1): Number of ways =

\frac{10 !}{5 ! 4 ! 1 !} = 1260

Total:

1260 + 1260 = 2520

JEE Advanced · 2026 · 17 May 2026 · Paper 1 · Numerical answer