How to check whether the value of binomial coefficient nCr is even or odd ?
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Well, you could always compute it, and look at the last digit. What kind of answer are you looking for? – Gerry Myerson May 30 '14 at 07:06
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4http://en.wikipedia.org/wiki/Lucas'_theorem#Consequence – complexist May 30 '14 at 07:07
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1Welcome to Math.SE! A more general result is described in this previous Answer, and the specific issue of parity of binomial coefficients is also previously asked. – hardmath May 30 '14 at 07:11
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Hint: You can use the Vandermonde convolution $\binom{n}{k}=\sum_{j=0}^k \binom{m}{j}\binom{n-m}{k-j}$ (for any $0\le m\le n$) for the case when $n$ is even (choose $m=n/2$) (you will notice that $\binom{n}{k}$ is even when $k$ is odd in this case). When $n$ is odd, you can use a similar strategy.
But, of course, as in the links provided in the comments, you may also use the more involved theorem due to Kummer.
mathse
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1Can you please say how do you deduce that $\binom{n}{k}$ is even when $n$ is even, $k$ odd from your formula? I am having trouble seeing it. – Asaf Shachar Aug 15 '17 at 19:29
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Maybe by example is best: $\binom{4}{1}=\binom{2}{0}\binom{2}{1}+\binom{2}{1}\binom{2}{0} = 2\binom{2}{1}\binom{2}{0}$. So it's always even. – mathse Aug 18 '17 at 09:29