The number of functions that maps an element in $A$ to another one in $A$ is $|A|^{|A|}$ for finite size of $A$, and the number of partial functions that maps an element in $A$ to another one in $A$ is $(1+|A|)^{|A|}$. If we divide the second by the first we get $$\frac{(1+|A|)^{|A|}}{|A|^{|A|}}=\left(\frac{1+|A|}{|A|}\right)^{|A|}=\left(1+\frac 1{|A|}\right)^{|A|}$$
whose limit is $e=2.71...$ as the size of $A$ goes to infinity, so by Cesaro lemma , in average, there $e$ times more partial functions that total functions. Is there a reason for this?