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In the spirit of the festive period and in appreciation of the encouraging response to my Xmas Combinatorics 2014 problem posted recently, here's one for the New Year!

Express the following as a product of four binomial coefficients: $$\color{red}{\sum_{r=1}^{M-1}r}\color{orange}{\sqrt{\int_0^X 2x dx}} \color{green}{\left(\prod_{i=1}^{120}(V!-i+1) \right)} \color{indigo}{\left(\prod_{h=1}^Mh\right)} \left[\color{maroon}{\left(\prod_{n=1}^{120}n\right)}\color{blue}{\left(\prod_{y=1}^M y\right)}\right]^{-1} $$

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$$\begin{align} &\color{red}{\sum_{r=1}^{M-1}r}{\color{orange}{\sqrt{\int_0^X 2x dx}}} \color{green}{\left(\prod_{i=1}^{120}(V!-i+1) \right)} \color{indigo}{\left(\prod_{h=1}^Mh\right)} \left[\color{maroon}{\left(\prod_{n=1}^{120}n\right)} \color{blue}{\left(\prod_{y=1}^M y\right)}\right]^{-1} \\\\ &=\color{orange}{\sqrt{\int_0^X 2x dx}} \color{red}{\sum_{r=1}^{M-1}r} \color{green}{\left(\prod_{i=1}^{120}(V!-i+1) \right)} \color{indigo}{\left(\prod_{h=1}^Mh\right)} \left[\color{maroon}{\left(\prod_{n=1}^{120}n\right)} \color{blue}{\left(\prod_{y=1}^M y\right)}\right]^{-1} \\\\ &=\color{orange}{\sqrt{\left[{x^2}\right]_0^X}}\cdot \color{red}{\frac{M(M-1)}2}\cdot \color{green}{V!^{\underline{120}}} \color{indigo}{M!} \left[\color{maroon}{120!}\color{blue}{M!}\right]^{-1} \\\\ &=\color{orange}X\cdot \color{red}{\frac{M(M-1)}2}\cdot \color{green}{V!^{\underline{120}}} \color{indigo}{M!} \left[\color{maroon}{120!}\color{blue}{M!}\right]^{-1} \\\\ &=\color{orange}{\frac X1}\cdot \color{red}{\frac{M(M-1)}{1\cdot 2}}\cdot \color{green}{\frac{V!^{\underline{120}}}{\color{maroon}{120!}}}\cdot \color{indigo}{\frac{M!}{\color{blue}{M!}}}\\\\ &=\color{orange}{\binom X1} \color{red}{\binom M2} \color{green}{\binom {V!}{120}} \color{indigo}{\binom MM}\\\\ &=\color{red}{\binom M2} \color{indigo}{\binom M0} \color{orange}{\binom X1} \color{green}{\binom {V!}{5!}}\\\\ \\\\ \end{align}$$

Happy New Year!

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  • 1
    $\begingroup$ Awesome!!${}{}{}{}$ $\endgroup$ – Dheeraj Kumar Jan 1 '15 at 15:37
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    $\begingroup$ $24 = 5!{}{}{}{}$? $\endgroup$ – peterwhy Jan 1 '15 at 15:37
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    $\begingroup$ I shall frame this one ! Thanks and Happy New Year !! $\endgroup$ – Claude Leibovici Jan 1 '15 at 15:40
  • $\begingroup$ @peterwhy - Thanks! Amended :) $\endgroup$ – hypergeometric Jan 1 '15 at 15:40
  • $\begingroup$ I saw the Merry X-mas one too !! :-) How do you think of these brilliant identities ? :D $\endgroup$ – sciona Jan 1 '15 at 15:44

protected by Alexander Gruber Jan 1 '15 at 22:29

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