JEE Advanced 2026 Paper 1Question 1
Let $S=\{1,2,3,\dots,10\}$. Consider the set $$X=\{R: R\text{ is an equivalence relation on the set }S\text{ such that }R\text{ has exactly }42\text{ elements}\}.$$ Then the number of elements in $X$ is ______________.
Sets, Relations and Function
Answer$2520$
Solution
An equivalence relation with blocks of sizes $n_i$ has $\sum n_i^2$ elements. $\sum n_i=10$, $\sum n_i^2=42$ only for $\{5,4,1\}$ and $\{6,2,1,1\}$: $\frac{10!}{5!4!1!}+\frac{10!}{6!2!1!1!\,2!}=1260+1260=2520$.