Pascal's Rule
\[{}^nC_r + {}^nC_{r+1}={}^{n+1}C_{r+1}\]
. Working on the left hand side:\[\begin{equation} \begin{aligned} \frac{n!}{r!(n-r)!}+\frac{n!}{(r+1)!(n-(r+1))!} &=\frac{n!}{r!(n-(r+1))!}(\frac{1}{n-r} + \frac{1}{r+1}) \\ &= \frac{n!}{r!(n-(r+1))!}\frac{n+11}{(n-r)(r+1)} \\ &= \frac{(n+1)!}{(r+1)(n-r)!} \\ &= \frac{(n+1)!}{(r+1)((n+1)-(r+1))!} \\ &= {}^{n+1}C_{r+1} \end{aligned} \end{equation}\]
