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 ________________.

<p>$\because$ $(B \cup C)' = B'\, \cap C'$</p> <p>B' is a set containing sub sets of A containing element 1 and not containing 2.</p> <p>And C' is a set containing subsets of A whose sum of elements is not prime.</p> <p>So, we need to calculate number of subsets of {3, 4, 5, 6, 7} whose sum of elements plus 1 is composite.</p> <p>Number of such 5 elements subset = 1</p> <p>Number of such 4 elements subset = 3 (except selecting 3 or 7)</p> <p>Number of such 3 elements subset = 6 (except selecting {3, 4, 5}, {3, 6, 7}, {4, 5, 7} or {5, 6, 7})</p> <p>Number of such 2 elements subset = 7 (except selecting {3, 7}, {4, 6}, {5, 7})</p> <p>Number of such 1 elements subset = 3 (except selecting {4} or {6})</p> <p>Number of such 0 elements subset = 1</p> <p>$n(B'\, \cap C') = 21 \Rightarrow n(B \cup C) = {2^7} - 21 = 107$</p>