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Math Help - combinatorics

  1. #1
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    combinatorics

    How do I go from to ?


    Thanks a lot!
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  2. #2
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    \sum_{i=0}^N\left(\begin{array}{c}N\\i\end{array}\  right)a^{i+1}b^{N-i}

    You shift the index of summation - instead of i running from 0 to N, we want it to run from 1 to N+1, so replace i with i-1:

    \sum_{i-1=0}^{i-1=N}\left(\begin{array}{c}N\\i-1\end{array}\right)a^{i-1+1}b^{N-(i-1)}

    and simplify:

    \sum_{i=1}^{N+1}\left(\begin{array}{c}N\\i-1\end{array}\right)a^ib^{N+1-i}

    Now bring out the N+1 term:

    \left(\begin{array}{c}N\\(N+1)-1\end{array}\right)a^{N+1}b^{N+1-(N+1)}+\sum_{i=1}^N\left(\begin{array}{c}N\\i-1\end{array}\right)a^ib^{N+1-i}

    which simplifies to:

    a^{N+1}+\sum_{i=1}^N\left(\begin{array}{c}N\\i-1\end{array}\right)a^ib^{N+1-i}

    - Hollywood
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