How many surjective functions from $A=\{1, 2, 3, 4\}$ to $B=\{a, b, c\}$ when $4 \mapsto c$?

Question

My thinking goes like this: since the $4$ from $A$ is already used, the problem can be rephrased as the number of surjective functions from $A_{new}=\{1,2,3\}$ to $B=\{a,b,c\}$. This can in turn be rephrased as the number of injective/bijective functions from $A_{new}$ to $B$ since the sets are of the same size.

This gives that the number of surjective functions from $A$ to $B$ is $3!=6$. However, the answer should be $12$. Why?

Answer 1

You need to count the functions where $f(\{1,2,3\})=\{a,b\}$ as well as $f(4)=c$. Of the eight functions from $\{1,2,3\}$ to $\{a,b\}$, there are two which are not surjective. This accounts for the six functions missing in your count.

Answer 2

Hint: $c$ is covered so you need to add up

• surjective functions $\{1,2,3\} \to \{a,b,c\}$
• surjective functions $\{1,2,3\} \to \{a,b\}$

These will be different functions, and there are six of each

Answer 3

You are missing functions from $\{1,2,3\}$ to $\{a,,b\}$ and there are six of them.

With your six functions you have the total twelve functions.