Prove that if p is prime and 1 less than or equal to k less than p then the binomial coefficient p = (p!)/(k!(p-k)!) is divisible by p.

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- Feb 10th 2008, 05:44 PMmandy123prove if p is prime then...
Prove that if p is prime and 1 less than or equal to k less than p then the binomial coefficient p = (p!)/(k!(p-k)!) is divisible by p.

k - Feb 10th 2008, 07:10 PMThePerfectHacker
This proves that for any $\displaystyle 1\leq k\leq p-1$ we have that $\displaystyle p(p-1)(p-2)...(p-k+1)$ is divisible by $\displaystyle k!$. Now $\displaystyle \gcd(k!,p) = 1$ because $\displaystyle p$ is a prime this means $\displaystyle (p-1)(p-2)...(p-k+1)$ is divisible by $\displaystyle k!$ thus $\displaystyle p(p-1)(p-2)...(p-k+1)$ is divisibly by $\displaystyle pk!$.