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Math Help - Pascals triangle!- questions

  1. #1
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    Pascals triangle!- questions

    Hi i have a few questions i had trouble with, i need some help! here are the questions,

    1) Find an equivalent nCr that is the sum of the following. The symmetric pattern. 44C19 + 44C20

    2)
    Find an equivalent nCr that is the sum of the following. This is the recursive pattern. 5C1 + 5C2

    Thanks in advance.
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    Re: Pascals triangle!- questions

    anyone ?
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    Re: Pascals triangle!- questions

    Quote Originally Posted by Gurp925 View Post
    Hi i have a few questions i had trouble with, i need some help! here are the questions,

    1) Find an equivalent nCr that is the sum of the following. The symmetric pattern. 44C19 + 44C20

    2)
    Find an equivalent nCr that is the sum of the following. This is the recursive pattern. 5C1 + 5C2

    Thanks in advance.
    I would be tempted to make use of the fact that \displaystyle \begin{align*} {n \choose{ r} } = \frac{n!}{r! \left( n - r \right) ! } \end{align*} and seeing what we get upon simplification of your expressions.
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    Re: Pascals triangle!- questions

    I should maybe tell you what the answer key says before i ask for help,

    1) 45C20

    2) 6C2
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    Re: Pascals triangle!- questions

    And what have you tried so far?
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    Re: Pascals triangle!- questions

    you can do it the following way

    nCr= \[\frac{{n!}}{{r!\left( {n - r} \right)!}}\]

    n C r+1= \[\frac{{n!}}{{(r + 1)!\left( {n - (r + 1)} \right)!}} = \frac{{n!}}{{(r + 1)r!\left( {n - (r)} \right)!/n - r}} = (n - r)/(r + 1)\frac{{n!}}{{r!\left( {n - r} \right)!}}\]

    therefore

    nCr + n C r+1 = \[\frac{{n!}}{{r!\left( {n - r} \right)!}} + \frac{{n!}}{{(r + 1)!\left( {n - (r + 1)} \right)!}} = (n - r)/(r + 1)\frac{{n!}}{{r!\left( {n - r} \right)!}} + \frac{{n!}}{{r!\left( {n - r} \right)!}} = \frac{{n!}}{{r!\left( {n - r} \right)!}}\left( {(n - r)/(r + 1) + 1} \right) = (n + 1)/(r + 1)\frac{{n!}}{{r!\left( {n - r} \right)!}}\]=n+1/r+1 nCr


    nCr + n C r+1= (n+1/r+1) nCr
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    Re: Pascals triangle!- questions

    Quote Originally Posted by Gurp925 View Post
    Hi i have a few questions i had trouble with, i need some help! here are the questions,
    1) Find an equivalent nCr that is the sum of the following. The symmetric pattern. 44C19 + 44C20
    2)
    Find an equivalent nCr that is the sum of the following. This is the recursive pattern. 5C1 + 5C2
    You are expected to know Pascals' equality: \binom{n+1}{k}=\binom{n}{k-1}+\binom{n}{k}
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