Let A = 1 , 2 , 3 , 4 , 5 , 6 , 7 . Define B = { T ⊆ A : either 1 ∉ T or 2 ∈ T } and C = {…

We know that,

By demorgan's law ${\left(B\cup C\right)}^{\text{'}}=B^{\text{'}}\cap C^{\text{'}}$

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

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

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

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

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

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

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

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

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

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

And number of elements in $n\left(B\cap C\right)=2^7-21=107$