Let $$A = \{1, 2, 3, 4, 5, 6, 7\}$$. Define $$B = \{T \subseteq A :$$ either $$1 \notin T$$ or $$2 \in T\}$$ and $$C = \{T \subseteq A :$$ the sum of all the elements of $$T$$ is a prime number $$\}$$. Then the number of elements in the set $$B \cup C$$ is ______.

Given,

$$A=\{1,2,3,4,5,6,7\}$$

and

$$B=\{T\subseteq A:\text{ either }1\notin T\text{ or }2\in T\}$$

Also,

$$C=\{T\subseteq A:\text{ sum of elements of }T\text{ is prime}\}$$

We need

$$n(B\cup C)$$

First find

$$n(B)$$

Total subsets of $$A$$ are

$$2^7=128$$

Now complement of $$B$$ occurs when

$$1\in T\quad \text{and}\quad 2\notin T$$

The remaining elements $$3,4,5,6,7$$ can be chosen freely.

Hence,

$$n(B')=2^5=32$$

Therefore,

$$n(B)=128-32=96$$

Now find subsets in $$B'$$ whose sum is prime.

For sets in $$B',$$

$$1\in T,\qquad 2\notin T$$

So write

$$T=\{1\}\cup S,$$

where

$$S\subseteq\{3,4,5,6,7\}$$

Now total sum becomes

$$1+\text{sum}(S)$$

Possible prime totals are

$$5,7,11,13,17,19,23,29$$

Hence required subset sums are

$$4,6,10,12,16,18,22,28$$

Now count them.

Sum $$4:$$

$$\{4\}\quad\Rightarrow1$$ case

Sum $$6:$$

$$\{6\}\quad\Rightarrow1$$ case

Sum $$10:$$

$$\{3,7\},\{4,6\}\quad\Rightarrow2$$ cases

Sum $$12:$$

$$\{5,7\}\quad\Rightarrow1$$ case

Sum $$16:$$

$$\{3,6,7\},\{4,5,7\}\quad\Rightarrow2$$ cases

Sum $$18:$$

$$\{5,6,7\}\quad\Rightarrow1$$ case

Sum $$22:$$

$$\{3,4,5,7\},\{4,5,6,7\}\quad\Rightarrow2$$ cases

Sum $$28:$$

$$\{3,4,5,6,7\}\quad\Rightarrow1$$ case

Therefore,

$$n(C\cap B')=1+1+2+1+2+1+2+1$$

$$=11$$

Hence,

$$n(B\cup C)=128-(32-11)$$

$$=128-21$$

$$=107$$

Therefore, the required number of elements is

$$\boxed{107}$$.