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*