Summing Up Binomial Coefficients

Asked Apr 17 '19 at 04:51

Active Apr 17 '19 at 07:33

Viewed 101 times

$$ \frac{\sum_{r=0}^{24}\binom{100}{4r}\binom{100}{4r+2}}{\sum_{r=1}^{25}\binom{200}{8r-6}} $$

I tried this problem using Complex number approach and arrived at the solution. When I tried the numerical 'normally' by summing up the binomial coefficients using various properties I was encountering some difficulty in simplifying it.

Can anybody give an alternate approach to this problem?

edited Apr 17 '19 at 07:33

Bill O'Haran

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asked Apr 17 '19 at 04:51

Ramone Estevez

Ramone, Sum of binomial coefficients is $2^n$, is this helpful? – sirous Apr 17 '19 at 05:18
Related: https://math.stackexchange.com/questions/3183307/to-find-the-sum-frac-1n-sum-binom-n23r-x1r/3183321#3183321 – lab bhattacharjee Apr 17 '19 at 05:19
I know that property but how to apply it here in the numerator ? – Ramone Estevez Apr 17 '19 at 05:22
For numerator, $$(1+x)^{100}(1+1/x)^{100}$$ – lab bhattacharjee Apr 17 '19 at 05:25
Can you please explain how ? – Ramone Estevez Apr 17 '19 at 05:27

Summing Up Binomial Coefficients

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