Let $[n]=\{ 1,2,\dots, n \}$ and $\binom{[n]}k$ denote the set of $k$-element subsets of $[n]$. Then $\binom{n}{k}=\#\binom{[n]}{k}$. > [!NOTE] Definition (Binomial Coefficient for Real Numbers) > Let $r\in \mathbb{R}$ and $k\in \mathbb{Z}.$ Then ${r \choose k}$ is given by ${r\choose k}=\begin{cases} \frac{r^{\underline{k}}}{k!}, & k \geq 0 \\ 0, & k<0 \end{cases}$where $r^{\underline{k}}$ denotes the [[Falling Factorial|falling factorial]]: that is when $r \geq 0,$ ${r \choose k} = \frac{r(r-1)\cdots(r-k+1)}{k(k-1)\cdots 1}$ # Properties By [[Factors of Binomial Coefficient]], $k{n \choose k }=n {n-1\choose k-1}.$ See [[Binomial Theorem]]. # Applications By [[Binomial Theorem]], .... By [[Number of k-Combinations of n Letters]], ...