I have two rows, on the first row, I have $A = \{1,2,3, \ldots, n\}$ and on the second row $B = \{n+1, n+2, \ldots, 2n\}$. What is the number of combinations of pairs that can be obtained with an element from every set?
For example if $A = \{1,2\}$ and $B = \{3,4\}$, then we have: $(1,3),(2,4)$ - first combination; $(1,4),(2,3)$ - the second combination. That mean for $n=2$ we have $2$ combinations.
Hint: You are really just trying to count the number of bijective functions $f\colon A \to B$, where $|A| = |B| = n$. The contents of each set aren't really important; you could pretend that $A = B$ so that you would be merely counting permutations.