12 Fold Way

Elements of $N$	Elements of $X$	Any $f$	$f$ injective (at most 1) $n\le X$	$f$ surjective (at least 1) $n\ge X$
Distinguishable	Distinguishable	$X^n$	$P(X,n)$	${\sum }_{k=0}^x\left(\genfrac{}{}{0ex}{}{X}{k}\right)(x-k{)}^n(-1{)}^k$
Indistinguishable	Ditinguishable	$\left(\genfrac{}{}{0ex}{}{n+X-1}{X-1}\right)$	$\left(\genfrac{}{}{0ex}{}{X}{n}\right)$	$\left(\genfrac{}{}{0ex}{}{n-1}{X-1}\right)$
Distinguishable	Indistinguishable	Sum of stirling numbers	1	Stirling numbers
Indistinguishable	Indistinguishable	partitions	1	partitions