JEE Main 2022 — Mathematics Probability & Statistics

Given,

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

So, total number of elements$=2022$=2022

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

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

Now let $A=$A=Number which are divisible by $2$2 from ${1,2,3,...2022}$1,2,3,...2022

So, $n(A)=1011$n(A)=1011

Now let $B=$B=Number which are divisible by $3$3by $3$3 from $(1,2,3,...2022)$(1,2,3,...2022)

So, $n(B)=674$n(B)=674

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

Now from ${1,2,3,...2022}$1,2,3,...2022 number multiple of $6$6 are $6,12,18,....2022$6,12,18,....2022

So, $n(A\cap B)=337$n(A∩B)=337

Now applying the formula we get, $n(A\cup B)=n(A)+n(B)-n(A\cap B)$n(A∪B)=n(A)+n(B)−n(A∩B)

$$=1011+674-337$$=1011+674−337

$$=1348$$=1348

Now let $C=$C=Number which divisible by $337$337 from ${1,.....1022}$1,.....1022

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

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

$$=2022-1350=672$$=2022−1350=672

Required probability$=\frac{672}{2022}=\frac{112}{337}$=2022672​=337112​