Permutations and combinations
The number of ways of selecting $$r$$ objects out of a total of $$n$$, where the order of selection is important, is the number of permutations: $$\displaystyle~^nP_r=\frac{n!}{(n-r)!}$$.
The number of ways in which $$r$$ objects can be selected from $$n$$ when the order of selection is not important is the number of combinations: $$\displaystyle~^nC_r =\left(\,\begin{array}{c}n\\r\\\end{array}\,\right)=\frac{n!}{r!(n-r)!}$$.
$$~^nC_n$$ must equal 1, so $$0!=1$$ and $$~^nC_0= 1$$; $$~^nC_r = ~^nC_{n-r}$$. Also \[~^nC_0 + ~^nC_1 +\ldots + ~^nC_n = 2^n\] \[~^{n+1}C_r = ~^nC_r + ~^nC_{r-1}\]