Xmas Maths 2014

Inspired the various** algebraic X’mas greetings sent to me over the festive period, I thought I would try to devise one of my own.

\Large \color{red}{\sum_{i=a-1}^{r-1}}\color{green}{\sum_{j=s-1}^{r-1}}\color{orange}{\binom {e-x}{m-x}}\color{red}{\binom ex}\color{orange}{ \binom i{a-1}}\color{green}{\binom j{s-1}}\color{red}{\binom y{\prod_{k=1}^{2014}k}}\\

The colours are purely ornamental!

** Actually there were only two versions: one was an equation with a \ln function and the other required knowledge of Newton’s second law; both of these have popped up in various places on web as well.

Answer

\large\begin{align}
& \color{red}{\sum_{i=a-1}^{r-1}}\color{green}{\sum_{j=s-1}^{r-1}}
\color{orange}{\binom {e-x}{m-x}}\color{red}{\binom ex}\color{orange}{ \binom i{a-1}}
\color{green}{\binom j{s-1}}\color{red}{\binom y{\prod_{k=1}^{2014}k}}\\
&=\color{orange}{\binom {e-x}{m-x}}\color{red}{\binom ex}\color{red}{\binom y{\prod_{k=1}^{2014}k}}\color{red}{\sum_{i=a-1}^{r-1}}
\color{orange}{ \binom i{a-1}}\color{green}{\sum_{j=s-1}^{r-1}}\color{green}{\binom j{s-1}}\\
&=\color{red}{\binom ex}\color{orange}{\binom {e-x}{m-x}}
\color{red}{\binom y{\prod_{k=1}^{2014}k}}\color{orange}{ \binom ra}\color{green}{\binom rs}\\
&=\color{red}{\binom em}\color{orange}{\binom mx}\color{red}{\binom y{2014!}}
\color{orange}{ \binom ra}\color{green}{\binom rs}\\
&=\color{orange}{\binom mx}\color{red}{\binom em}\color{orange}{ \binom ra}
\color{green}{\binom rs}\color{red}{\binom y{2014!}}
\end{align}

Merry Xmas, everyone!!!

Attribution
Source : Link , Question Author : Hypergeometricx , Answer Author : Hypergeometricx

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