+26376 Simplify C(n,k)/C(n,k-1)
\(\begin{array}{|rcll|} \hline \dfrac{C(n,k)} {C(n,k-1)} &=& \dfrac{ \dbinom{n}{k} }{ \dbinom{n}{k - 1} } \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \mathbf{\dbinom{n}{k - 1}} &=& \dfrac{n!}{(k-1)!(n-k+1)!} \quad &| \quad (k-1)!k=k!~ \text{or}~(k-1)!= \dfrac{k!}{k} \\\\ &=& k *\dfrac{n!}{k!(n-k+1)!} \quad &| \quad (n-k)!(n-k+1)=(n-k+1)! \\\\ &=& \dfrac{k}{n-k+1} * \dfrac{n!}{k!(n-k)!} \quad &| \quad \dfrac{n!}{k!(n-k)!} = \dbinom{n}{k} \\\\ &=& \mathbf{ \dfrac{k}{n-k+1} \dbinom{n}{k} } \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \dfrac{ \dbinom{n}{k} }{ \dbinom{n}{k - 1} } &=& \dfrac{ \dbinom{n}{k} }{ \dfrac{k}{n-k+1} \dbinom{n}{k} } \\\\ \dfrac{ \dbinom{n}{k} }{ \dbinom{n}{k - 1} } &=& \dfrac{1}{ \dfrac{k}{n-k+1} } \\\\ \mathbf{\dfrac{ \dbinom{n}{k} }{ \dbinom{n}{k - 1} } } &=& \mathbf{\dfrac{n-k+1}{k}} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \mathbf{\dfrac{C(n,k)} {C(n,k-1)}} &=& \mathbf{\dfrac{n-k+1}{k}} \\ \hline \end{array}\)
