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Math Help - Combinatorics

  1. #1
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    Combinatorics

    Dear Colleagues,

    Could you please help me in solving the following problem:
    What is the value of \binom {20}{0} +\binom {20}{2} +\binom {20}{4} +... \binom {20}{20} ?


    Best Regards.
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  2. #2
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    Quote Originally Posted by raed View Post
    Dear Colleagues,

    Could you please help me in solving the following problem:
    What is the value of \binom {20}{0} +\binom {20}{2} +\binom {20}{4} +... \binom {20}{20} ?


    Best Regards.
    Hint:

    Consider


    (1+1)^{20}

    Use the binomial theorem
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  3. #3
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    Quote Originally Posted by raed View Post
    Could you please help me in solving the following problem: What is the value of \binom {20}{0} +\binom {20}{2} +\binom {20}{4} +... \binom {20}{20} ?
    \left( {x + y} \right)^n  = \sum\limits_{k = 0}^n {\binom{n}{k}x^k y^{n - k} }.
    What if x=1~\&~y=1~?
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  4. #4
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    Thank you very much for your reply. In fact I tried that but what about terms \binom {20}{1}, \binom {20}{3},...,\binom {20}{19} ?


    Best Regards.
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  5. #5
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    Quote Originally Posted by raed View Post
    Thank you very much for your reply. In fact I tried that but what about terms \binom {20}{1}, \binom {20}{3},...,\binom {20}{19} ?
    x=1~\&~y=-1 then what?
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