JEE Main · 2022 · 29 Jul 2022 · Shift 1 · Probability

First, factorize $2022=2\times 3\times 337$. We need numbers in $\{1,2,\dots ,2022\}$ that are coprime to 2022, i.e., not divisible by 2, 3, or 337.

Let A = numbers divisible by 2: $n(A)=1011$.
Let B = numbers divisible by 3: $n(B)=674$.
Let C = numbers divisible by 337: $n(C)=6$.

$n(A\cap B)$ = numbers divisible by 6 = 337.
$n(A\cap C)$ = numbers divisible by 674 = 3.
$n(B\cap C)$ = numbers divisible by 1011 = 2.
$n(A\cap B\cap C)$ = numbers divisible by 2022 = 1.

By inclusion-exclusion:


Numbers coprime to 2022: $2022-1350=672$.

Probability = $\frac{672}{2022}=\frac{112}{337}$.