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Let $A :\{1,2,3,4,5,6,7\}$. Define $B =\{ T \subseteq A$ : either $1 \notin T$ or $2 \in T \}$ and $C = \{ T \subseteq A : T$ the sum of all the elements of $T$ is a prime number $\}$. Then the number of elements in the set $B \cup C$ is $\dots\dots$
$107$
$106$
$105$
$108$
Solution
$B^{\prime}$ is a set containing sub sets of $A$ containing element $1$ and not containing $2$ .
And $C^{\prime}$ is a set containing subsets of $A$ whose sum of elements is not prime.
So, we need to calculate number of subsets of $\{3,4,5,6,7\}$ whose sum of elements plus $1$ is composite.
Number of such $54\,elements\,subset\,=1$
Number of such $4$ elements subset $=3$ (except selecting $3$ or $7$ )
Number of such $3$ elements subset $=6$ (except selecting $\{3,4,5\},\{3,6,7\},\{4,5,7\}$ or $\{5,6,7\}$ )
Number of such $2$ elements subset $=7$ (except selecting $\{3,7\},\{4,6\},\{5,7\})$
Number of such $1$ elements subset $=3$ (except selecting $\{4\}$ or $\{6\}$ )
Number of such $0$ elements subset $=1$
$n\left(B^{\prime} \cap C^{\prime}\right)=21 \Rightarrow n(B \cup C)=2^{7}-21=107$
