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Math Help - Pascal's Rule

  1. #1
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    Pascal's Rule

    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. A member kindly said it was this rule but I don't know how to prove
    it for this question using the identity:

    (n , r-1)+(n ,r)=(n+1, r) written as column vectors

    Thanks
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  2. #2
    MHF Contributor undefined's Avatar
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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. A member kindly said it was this rule but I don't know how to prove
    it for this question using the identity:

    (n , r-1)+(n ,r)=(n+1, r) written as column vectors

    Thanks
    I see you are unsatisfied by my responses in this thread. Why not just ask about what you didn't understand?
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Pascal's Rule

    Combinatorial proof:

    Pascal's Rule states:

    {n\choose k} = {n-1\choose k} + {n-1 \choose k-1}

    Proof:


    Let x be an element of group of n elements.
    The number of combinations of k elements from group of n elements, which is not containing the element x is give by: {n-1\choose k}, now, the number of combinations which containing element x is: {n-1 \choose k-1}.

    But, every combination is containing x or not containing x. Hence, the total number {n\choose k} of combinations of k from n elements is the sum of the above.

    Q.E.D
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