Arifa has learned about mapping recently. She has defined a map by her name such that the map goes from $\{1, 2, 3, \dots , n \}$ to $\{1, 2, 3, \dots , n \}$ such that every distinct element maps to another distinct element. How many possible maps are there for $n=10$ such that there exists at least one $i$ such  that $Arifa(i) = i$.