05A10-PascalsRuleProof.tex

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\pmtitle{Pascal's rule proof}
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\usepackage{amssymb}
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\begin{document}
We need to show 
\begin{eqnarray*}
\binom{n}{k} + \binom{n}{k-1} & = & \binom{n+1}{k}
\end{eqnarray*}
Let us begin by writing the left-hand side as $$ \frac{n!}{k!(n-k)!} + \frac{n!}{(k-1)!(n-(k-1))!}$$
Getting a common denominator and simplifying, we have 
\begin{eqnarray*}
\frac{n!}{k!(n-k)!} + \frac{n!}{(k-1)!(n-k+1)!} & = & \frac{(n-k+1)n!}{(n-k+1)k!(n-k)!}+\frac{kn!}{k(k-1)!(n-k+1)!} \\ 
& = & \frac{(n-k+1)n!+kn!}{k!(n-k+1)!} \\
& = & \frac{(n+1)n!}{k!((n+1)-k)!} \\
& = & \frac{(n+1)!}{k!((n+1)-k)!} \\
& = & \binom{n+1}{k}
\end{eqnarray*}
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