Evenness of binomial co coefficients for power $2^n$.

Asked Nov 10 '17 at 09:43

Active Nov 10 '17 at 10:28

Viewed 77 times

I've noticed that all binomial coefficients, apart from first and last, for powers 1, 2, 4 and 8 are even numbers. This suggests, that it may be the case for any $2^n$.

In other words, is the following true?

$\binom{2^n}{k}$ is even for any natural $k<2^n$

edited Nov 10 '17 at 10:28

asked Nov 10 '17 at 09:43

cyanide

1

You can go further: Given ay prime $p$, $p$ divides $\binom{p^n}{m}$ for any $m, n$ with $1\leq m\leq p^n-1$. It all comes down to the number of factors $p$ in a given factorial. Do you know how to tell that number? – Arthur Nov 10 '17 at 09:48
Yes. ${{{{}}}}$ – Nov 10 '17 at 09:49
Can you prove it, or give a link to the proof? – cyanide Nov 10 '17 at 09:53
@cyanide I realized now that your question is already answered here: https://math.stackexchange.com/questions/317163/prove-if-n-2k-1-then-binomni-is-odd-for-0-leq-i-leq-n – Robert Z Nov 10 '17 at 10:40
2

Special case of Prime dividing the binomial coefficients – Peter Taylor Nov 10 '17 at 11:10

Evenness of binomial co coefficients for power $2^n$.

0 Answers0