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Math Help - Is this number even or odd?

  1. #1
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    Is this number even or odd?

    Decide if the binomial coefficient  {100 \choose 68} is even or odd.
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  2. #2
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    Quote Originally Posted by Intsecxtanx View Post
    Decide if the binomial coefficient  {100 \choose 68} is even or odd.


    We know (and if not prove it!) that the maximal power of a prime p dividing n! is given by \sum^\infty_{k=1}\left[n/p^k\right] , where [x] is the floor function = the greatest integer that is less than or equal x.

    Note that the above sum is finite since \left[n/p^r\right]=0\,\,\,for\,\,\,p^r>n

    So the maximal power of 2 dividing 100! is 50+25+12+6+3+1=97 , and the max. power of 2 dividing 32! is 16+8+4+2+1=31 , and the one

    dividing 68 is 34+17+8+4+2+1=66 , thus the number is odd.

    Tonio
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