Guided Example #2 - Natural Numbers

${\displaystyle \left(\begin{array}{c}m\\ k\end{array}\right)+\left(\begin{array}{c}m\\ k-1\end{array}\right)=\frac{m!}{k!(m-k)!}+\frac{m!}{(k-1)!(m-k+1)!}}$

${\displaystyle =\frac{m!}{k(k-1)!(m-k)!}+\frac{m!}{(k-1)!(m-k+1)(m-k)!}}$

${\displaystyle =\frac{m!(m-k+1)+km!}{k(k-1)!(m-k+1)(m-k)!}}$

${\displaystyle \frac{m!m+m!}{k(k-1)!(m-k+1)(m-k)!}}$

${\displaystyle \frac{m!(m+1)}{k!(m-k+1)!}}$

${\displaystyle \frac{(m+1)!}{k!(m+1-k)!}}$

${\displaystyle \left(\begin{array}{c}m+1\\ k\end{array}\right)}$

${\displaystyle =\frac{m!}{k(k-1)!(m-k)!}+\frac{m!}{(k-1)!(m-k+1)(m-k)!}}$

${\displaystyle =\frac{m!(m-k+1)+km!}{k(k-1)!(m-k+1)(m-k)!}}$

${\displaystyle \frac{m!m+m!}{k(k-1)!(m-k+1)(m-k)!}}$

${\displaystyle \frac{m!(m+1)}{k!(m-k+1)!}}$

${\displaystyle \frac{(m+1)!}{k!(m+1-k)!}}$

${\displaystyle \left(\begin{array}{c}m+1\\ k\end{array}\right)}$