The number of distinct partial binary operations on a finite set of n elements

Question

I am asked to show that there are exactly $(n+1)^{n^2}$ partial binary operations on a finite set of n elements.

My professor said that this can be done using a combinatoric argument, but I have failed to see how.

Things I know: There are exactly $n^{n^2}$ different binary operations on any given finite set of n elements. Also every binary operation is a partial binary operation, but not vice versa. Therefore there should be more partial binary operations than there are plain binary operations.

Any hints or clues would be much appreciated. Thank you for your time.

DEFINITION: A binary operation on a set $S$ is a function from $S \times S$ to $S$.

Answer 1

HINT: Add an $(n+1)$-st value, $\text{undefined}$, to the set of possible values of the operation.