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Math Help - Binomial Number proof

  1. #1
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    Binomial Number proof

    Hi,

    I'm required to show that

    {n \choose r} + 2{n \choose r-1} + {n \choose r-2} = {n+1 \choose r}

    Now I know that {n \choose r} + {n \choose r-1} = {n+1 \choose r}

    and we can write the LHS as

    \left[{n \choose r} + {n \choose r-1}\right] + {n \choose r-1} + {n \choose r-2}

    ={n+1 \choose r} + {n \choose r-1} + {n \choose r-2}

    Which means for the equality to hold I would have to show that {n \choose r-1} + {n \choose r-2} = 0, which I don't think is right.

    So where have I messed up?

    Thanks
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  2. #2
    Moo
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    Hello,

    I'd say it's rather {n \choose r} + 2{n \choose r-1} + {n \choose r-2} = {n+{\color{red}2} \choose r}

    Because using the formula, we have {n \choose r-1} + {n \choose r-2}={n+1 \choose r-1}

    And using the formula again, it results that :
    {n \choose r} + 2{n \choose r-1} + {n \choose r-2}={n+1 \choose r} +{n+1 \choose r-1}={n+2 \choose r}

    So could there possibly be a typo ?
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  3. #3
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    Quote Originally Posted by Moo View Post
    So could there possibly be a typo ?
    Double checked and I definitely copied the question out correctly, so I guess it must be a typo.

    Thanks
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