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.

Solution 1:

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

Merry Xmas, everyone!!!