JEE Main 2022 — Mathematics Algebra

We know that,

By demorgan's law ${(B\cup C)}^{'}={B}^{'}\cap {C}^{'}$(B∪C)′=B′∩C′

${B}^{'}$B′ is a set containing sub sets of $A$A containing element $1$1 and not containing $2$2.

And ${C}^{'}$C′ is a set containing subsets of $A$A whose sum of elements is not prime.

So, we need to calculate number of subsets of ${3,4,5,6,7}$3,4,5,6,7 whose sum of elements plus $1$1 is composite.

Number of such $5$5 elements subset $=1$=1

Number of such $4$4 elements subset $=3$=3 (except selecting $3$3 or $7$7)

Number of such $3$3 elements subset $=6$=6 (except selecting ${3,4,5},{3,6,7},{4,5,7}$3,4,5,3,6,7,4,5,7 or ${5,6,7}$5,6,7)

Number of such $2$2 elements subset $=7$=7 (except selecting ${3,7},{4,6},{5,7}$3,7,4,6,5,7)

Number of such $1$1 elements subset $=3$=3 (except selecting ${4}$4 or ${6}$6)

Number of such $0$0 elements subset $=1$=1

So, number of elements in $n({B}^{'}\cap {C}^{'})=1+3+6+7+3+1=21$n(B′∩C′)=1+3+6+7+3+1=21

And number of elements in $n(B\cap C)={2}^{7}-21=107$n(B∩C)=27−21=107