For any finite set $X,$  let $\left|X\right|$  denote the number of elements in $X,$ . Define $S_n=\sum \left|A\cap B\right|,$
 where the sum is taken over all ordered pairs  $\left(A,B\right)$ such that$A$   and $B$  are subsets of$\left\{1,2,3,\mathrm{.......},n\right\}$  with$\left|A\right|=\left|B\right|$ . 
(For example,  $S_2=4$ because the sum is taken over the pairs of subsets$\left(A,B\right)\in \left\{\left(0,0\right),\left(\left\{1\right\},\left\{1\right\}\right)\left(\left\{1\right\},\left\{2\right\}\right),\left(\left\{2\right\},\left\{1\right\}\right),\left(\left\{2\right\},\left\{2\right\}\right),\left(\left\{1,2\right\},\left\{1,2\right\}\right)\right\},$$\Rightarrow S_2=0+1+0+0+1+2=4.$
  ) Then $S_3-10=$