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Please help me evaluate the following Binomial coefficient using any known properties of Pascal triangle: $$ \binom{52}{4} - \binom{47}{4} + \binom{47}{5} $$

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Do you have the formula $${n \choose m}=\frac{n!}{m!(n-m)!}~?$$ – anon Apr 19 '12 at 7:48
Well, what are the facts about Pascal triangle, you know, what have you tried with them...? @JSSumantri – user21436 Apr 19 '12 at 8:23
The original problem is $$\binom{47}{5} + \sum_{i=1}^{5}\binom{52-i}{3}$$ Is there any simpler / more elegant way to evaluate the above expression rather than using the factorial formula? – JS SUMANTRI Apr 19 '12 at 8:38
@JSSUMANTRI Add the comment to your question, please. – user21436 Apr 19 '12 at 9:23

I think at this point we can safely say that there is no method for evaluating the quantity in the display that is any better than the obvious method. By "the obvious method," I mean writing it as $${(52)(51)(50)(49)\over24}-{(47)(46)(45)(44)\over24}+{(47)(46)(45)(44)(43)\over120}$$ and then doing the arithmetic. Maybe the computational burden is lighter if you put everything over 120 and do some distributing: $${(52)(51)(50)(49)(5)+(47)(46)(45)(44)(43-5)\over120}$$ Or maybe it's simpler to do what cancellations you can to get $$(13)(17)(25)(49)-(47)(23)(15)(11)+(47)(23)(3)(11)(43)$$ Whatever, at the end of the day you have some arithmetic to do.

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