Binomial coefficient modulo a prime

  1. A question:

    Let bin(a,b) denote the binomial coefficient a! / ( b! (a - b)! ).

    Is it true that

    bin( 2p, p ) = 2 (mod p) if p is prime and p>=3 ?
  2. jcsd
  3. matt grime

    matt grime 9,395
    Science Advisor
    Homework Helper

    Yes, it's fermat's little theorem: x^p=x mod p, for p a prime, hence

    (1+x)^2p = (1+x^p)^2 = 1+2x^p+x^{2p} mod p

    note your requirement on p>=3 is not necessary. 4 choose 2 =6 whcih is congruent to 2 mod 2 as well.
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