Let $A=\left\{1,2,3,4,5,6,7\right\}$. 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 _______.

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 $n\left(B\cap C\right)=2^7-21=107$