# Math Help - Binomial Theorem - Prove that result is valid only for...?

1. ## Binomial Theorem - Prove that result is valid only for...?

Prove that $\left({m\choose 0} + {m\choose 1} - {m\choose 2} - {m\choose 3}\right) + \left({m\choose 4} + {m\choose 5} - {m\choose 6} - {m\choose 7}\right) + \mbox{...} = 0$ if and only if for some positive integer $k,\ m = 4k - 1$

2. try it for small m, and use that (m n) = (m m-n) should be pretty obvious that if the condition doesn't hold the series doesn't 'telescope' into itself and leaves a remainder.
try the telescoping for small m if u want to see what's happening.

3. $\binom{n-1}{r-1}+\binom{n-1}{r}=\binom{n}{r}
$

4. no real need for that, just need that (7 0) = (7 7), (7 1) = (7 6) etc.