$A$ and $B$ are two sets and their intersection has $3$ elements. If $A$ has $1920$ more subsets than $B$ has, then the number of elements of $A$ union $B$ is:

$$\begin{gathered} A=\left\{a_1{a}_2.......a_n p_1{p}_2{p}_3\right\} \\ B=\left\{b_1{b}_2......b_m p_1{p}_2{p}_3\right\} \\ {2}^{n+3}=2^{m+3}+1920\Rightarrow 2^n-2^m=240 \\ =2^m\left\{2^{n-m}-1\right\}=240=15\times 16=15\times 2^4 \\ \therefore 2^m=2^4\Rightarrow m=4, 2^{n-m}=16 \\ n-m=4 \\ \therefore n=8 \\ n\left(A\cup B\right)=8+4+3=15 \end{gathered}$$