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Math Help - 2 doubts ....help!

  1. #1
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    2 doubts ....help!

    1. if  a^2+b^2 = c^2 ,please show why  (a+b) and  c have the same parity?

    2. proof this using a combinatorial argument:
     \sum\limits_{r=0}^{n}C_r^n 2^{r} = 3^{n}
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  2. #2
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    2. Hint: \displaystyle \sum_{r=0}^{n}\binom{n}{r}2^r =  \sum_{r=0}^{n}\binom{n}{r}(1)^{n-r}2^r.
    Last edited by TheCoffeeMachine; November 18th 2010 at 02:31 AM.
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  3. #3
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    1) (a+ b)^2= a^2+ 2ab+ b^2
    "2ab" is clearly even so the parity of (a+ b)^2 is the same as the partity of c^2= a^2+ b^2.

    And, of course, since (2n)^2= 4n^2 and (2n+ 1)^2= 4n^2+ 4n+ 1= 2(2n^2+ 2n)+ 1 the parity of a^2 is the same as the parity of a.
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  4. #4
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    @TheCoffeeMachine
    yes thanks!
    but i want a proof by a combinatorial argument.
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  5. #5
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    Well, you can prove the binomial theorem (which is what TheCoffeeMachine used) with a combinatorial argument.
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