Let S = \ 1, 2, 3, , 2022\ . Then the probability, that a randomly...

Given,

$S = \left\{1, 2, 3, \ldots , 2022\right\}$

So, total number of elements$= 2022$

Now factors of $2022 = 2 \times 3 \times 337$

Now $HCF \left(n, 2022\right) = 1$ is feasible only when the value of '$n$' and $2022$ has no common factor.

Now let $A =$Number which are divisible by $2$ from \left\{1, 2, 3, . . . 2022\right\}$</p><p>So,$n\left(A\right) = 1011$</p><p>Now let$B =$Number which are divisible by$3$by$3$from \left(1, 2, 3, . . . 2022\right)$

So, $n\left(B\right) = 674$

Now $A \cap B =$Number which are divisible by $6$

Now from \left\{1, 2, 3, . . . 2022\right\}$number multiple of$6$are$6, 12, 18, . . . . 2022$</p><p>So,$n\left(A \cap B\right) = 337$</p><p>Now applying the formula we get,$n\left(A \cup B\right) = n\left(A\right) + n\left(B\right) - n\left(A \cap B\right)$</p><p>$= 1011 + 674 - 337$</p><p>$= 1348$</p><p>Now let$C =$Number which divisible by$337$from \left\{1, . . . . . 1022\right\}$

Total elements which are divisible by $2$ or $3$ or $337 = 1348 + 2 = 1350$

Favourable cases=elements which are neither divisible by $2, 3$ or $337$

$= 2022 - 1350 = 672$

Required probability$= \dfrac{672}{2022} = \dfrac{112}{337}$