# Help with surjective maps

## Homework Statement

Let S = {1,2,3,...,n}

How many surjective maps are there from S to S?

n/a

## The Attempt at a Solution

However, I thought that total number of surjective maps = n^n because 1-1 isn't required. Where am I wrong?

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hunt_mat
Homework Helper
This is understandable as a surjective map from itself to itself, and this is essential just the set S being rearranged, so the answer will be n!.

Mat

1-1 is implicitly required, because each element of the domain can only map to one element of the range - so to map to every element of the range you'll need a 1-1 mapping.

LCKurtz
Homework Helper
Gold Member

## Homework Statement

Let S = {1,2,3,...,n}

How many surjective maps are there from S to S?