Please help me evaluate the following Binomial coefficient using any known properties of Pascal triangle:
$$
\binom{52}{4} – \binom{47}{4} + \binom{47}{5}
$$
[Math] Evaluate the Binomial Coefficient
binomial-coefficients
binomial-coefficients
Please help me evaluate the following Binomial coefficient using any known properties of Pascal triangle:
$$
\binom{52}{4} – \binom{47}{4} + \binom{47}{5}
$$
Best Answer
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.