The following information could be useful:
The twelvefold way
R.P. Stanley presents in his classic Enumerative combinatorics vol. 1 in section 1.9 the so-called twelvefold way. He considers finite sets $N$ and $X$, with $|N|=n, |X|=x$ and counts the number of different functions $f:N\rightarrow X$ under different situations.
Functions: $f$ may be arbitrary, injective, surjective giving three different possibilities.
Sets: Elements of $N,X$ may be either distinguishible or indistinguishible resulting in four different possibilities.
Altogether we can consider $3\cdot 4=12$ different situations:
\begin{array}{llccc}
\text{Elements}&\text{Elements}\\
\text{of }N&\text{of }X&\quad\text{Any }f\quad&\quad\text{Injective }f\quad&\quad\text{Surjective } f\quad\\
\hline
\text{dist.}&\text{dist.}&x^n\quad&\quad x^{\underline{n}}\quad&x!{n\brace x}\\
\text{indist.}&\text{dist.}&\left(\!\!{x\choose n}\!\!\right)\quad&\quad\binom{x}{n}\quad&\left(\!\!{x\choose n-x}\!\!\right)\quad\\
\text{dist.}&\text{indist.}&\sum_{j=0}^x{n\brace j}\quad&\quad\begin{matrix}1&\text{if }n\leq x\\0&\text{if }n>x\end{matrix}\quad&{n\brace x}\quad\\
\text{indist.}&\text{indist.}&\sum_{j=0}^xp_j(n)\quad&\quad\begin{matrix}1&\text{if }n\leq x\\0&\text{if }n>x\end{matrix}\quad&p_x(n)\quad\\
\end{array}
with
$x^{\underline{n}}=x(x-1)\cdots(x-n+1)$ the falling factorial power,
${n\brace x}$ the Stirling numbers of second kind and
$p_x(n)$ the number of partitions of $n$ into $x$ parts.
I recommend a thorough study of this instructive section.
