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Math Help - Combination: Proof

  1. #1
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    Combination: Proof

    Hey all!

    I have a question, which I've been pondering over for a few hours, and just can't seem to grasp it.

    It goes:

    (m+n)C(3) = (m)C(3) + (m)C(2)*(n)C(1) + (m)C(1)*(n)C(2) + (n)C(3)

    I've tried simplify the LHS down to (m+n)(m+n-1)(m+n-2)/3! But this didn't help, and I've tried simplify, and canceling down the RHS too, but I always end up with a huge expansion (18 terms ).

    Could anyone lend a helping hand? Or some insight into the problem?

    Thanks!
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  2. #2
    MHF Contributor red_dog's Avatar
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    We have (1+x)^{m+n}=(1+x)^m(1+x)^n

    The coefficient of x^3 in the LHS is C_{m+n}^3

    The coefficient of x^3 in the RHS is C_m^3+C_m^2C_n^1+C_m^1C_n^2+C_n^3

    But the coefficients are the same, so we have the equality.
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  3. #3
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    Hello, and thank you.

    However, could you just clarify the first line? I don't comprehend how you structure? or deduced it.
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  4. #4
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    Quote Originally Posted by Khorne View Post
    could you just clarify the first line? I don't comprehend how you structure? or deduced it.
    You do know the basic laws of exponents.
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  5. #5
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    Indeed, but not with this type of proof setting.
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