If$A=\{a,b,c,d\}$ , the number of anti-symmetric relations $2^n\times 3^m$, then $m+n=$

$A=\{a,b,c,d\}\Rightarrow n\left(A\right)=4$

Number of anti-symmetric relations $=2^n\cdot 3^{\frac{n(n-1)}{2}}$

$=2^4\times 3^{\frac{4\times 3}{2}}=2^4\times 3^6$

Comparing $2^m\times 3^n\Rightarrow m=4;n=6\Rightarrow m+n=10$