Fix the kingdoms and permute 2n champions$\implies (2n)!\; ways$
In this $(2n)!$ every kingdom counted its champions twice. i.e. $(C_i,C_j)$ and $(C_j,C_i)$ both are counted..
So we should have to divide (2n)! by 2 for each kingdom. And there are n kingdoms.
$\implies \frac{(2n)!}{2^n}$