This question concerns functions $f:\{A,B,C,D,E\}\rightarrow\{1,2,3,4,5,6,7\}$ (counting) Can someone guide me towards a way to count surjective functions of the below question?

This question concerns functions $f:\{A,B,C,D,E\}\rightarrow\{1,2,3,4,5,6,7\}$. How many such functions are there? How many are injective? Surjective? Bijective?

My answers with logic behind them:
There are a total of $7^5$ functions since each $f(k)$ where $k\in\{A,B,C,D,E\}$ may map to $7$ elements in the set $\{1,2,3,4,5,6,7\}$.
The number of injective functions is $7\cdot6\cdot5\cdot4\cdot3$ since once we select an element to map to we may not map to it again since injectivity means that if $x\neq y\Rightarrow f(x)\neq f(y)$.
Not sure on surjective count... 

There are no surjective functions from a finite set to a bigger finite one.
