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.

  • 6
    $\begingroup$ Nice ! (+1) Merry Christmas and good-luck with it :) $\endgroup$ – r9m Dec 26 '14 at 9:34
  • 4
    $\begingroup$ @r9m - Thanks and Merry Xmas! $\endgroup$ – hypergeometric Dec 26 '14 at 9:45
  • $\begingroup$ Should that first sum be from $i = a-1$ to $r-1$ instead of $r=1$? $\endgroup$ – JimmyK4542 Dec 26 '14 at 9:54
  • $\begingroup$ @JimmyK4542 - Yes, that's right - thanks! Amended. $\endgroup$ – hypergeometric Dec 26 '14 at 10:01
  • 3
    $\begingroup$ Why put everything in \large and even in \Large? $\endgroup$ – Did Dec 26 '14 at 11:29

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

Merry Xmas, everyone!!!

  • 9
    $\begingroup$ It's times like this that I realise I don't have a clue what's going on and that I really don't belong here. $\endgroup$ – Pharap Dec 27 '14 at 12:12
  • 1
    $\begingroup$ @Pharap - You just have to read the last line of the equation! :) $\endgroup$ – hypergeometric Jan 1 '15 at 15:37

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