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Math Help - Binomial Theorem proof

  1. #1
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    Binomial Theorem proof

    Hi this question was in Advanced Higher Maths in the UK
    Show that (n+1 , 3) - (n ,3 )=(n ,2)
    They are written in column vector form ie n+1 at top and 3 at the bottom for example for the first. I think you have to use the n c r formula to prove the rhs
    Please Help
    Thanks
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  2. #2
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    Quote Originally Posted by minicooper58 View Post
    Hi this question was in Advanced Higher Maths in the UK
    Show that (n+1 , 3) - (n ,3 )=(n ,2)
    They are written in column vector form ie n+1 at top and 3 at the bottom for example for the first. I think you have to use the n c r formula to prove the rhs
    Please Help
    Thanks
    This is a special case of Pascal's rule.
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  3. #3
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    hi again the following link takes you to the question http://i223.photobucket.com/albums/d...f/HPIM0534.jpg
    thanks
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  4. #4
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    Many thanks for the Pascal's Rule. Perfect.
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  5. #5
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    Hi, I understand that Pascal's rule is involved but I'm not sure how to go about the proof. The link to the question no 5 is at the following link http://i223.photobucket.com/albums/d...f/HPIM0534.jpg. It just seems to easy to simplify and make r =3.
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  6. #6
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    Quote Originally Posted by minicooper58 View Post
    It just seems to easy to simplify and make r =3.
    Well if you don't want to use Pascal's rule, try this

    \displaystyle \binom{n+1}{3} - \binom{n}{3}

    \displaystyle =\frac{(n+1)!}{3!(n-2)!} - \frac{n!}{3!(n-3)!}

    \displaystyle =\frac{(n+1)!-n!(n-2)}{3!(n-2)!}

    \displaystyle =\frac{(n+1)(n!)-(n!)(n-2)}{3!(n-2)!}

    \displaystyle =\frac{(n!)(n+1-(n-2))}{3!(n-2)!}

    \displaystyle =\frac{3n!}{3!(n-2)!}

    \displaystyle =\frac{n!}{2!(n-2)!}

    \displaystyle =\binom{n}{2}
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