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Combinatorics Class - Sum Question

  1. Sep 17, 2012 #1
    1. The problem statement, all variables and given/known data
    For any positive integer n determine:

    [itex]\sum\limits^n_{i=0} \frac{1}{i!(n-i)!}[/itex]


    2. Relevant equations

    I don't really know where to start.. Up until this point we've just been doing permutations, combinations, and determining the coefficient of a certain term in the expansion of a polynomial. There aren't any examples like this question in the text, and so I am unsure as to what sort of an answer they are looking for... Are they just looking for a general formula (not a sum) for the answer, with n as a variable? Cheers for any direction!!


    3. The attempt at a solution
     
  2. jcsd
  3. Sep 17, 2012 #2

    jbunniii

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    Hint: does this look familiar?

    [tex]\frac{n!}{i!(n-i)!}[/tex]
     
  4. Sep 17, 2012 #3
    So the answer I'm looking for is

    [itex]\frac{\dbinom{n}{i}}{n!}[/itex]

    Correct?
     
  5. Sep 17, 2012 #4
    Or will it be

    [itex]\sum\limits^n_{i=0} \dfrac{\dbinom{n}{i}}{n!}[/itex]

    I'm confused as to whether the sum is still involved.
     
  6. Sep 17, 2012 #5
    you should find the following sum:

    [itex] \frac{1}{n!}*\sum \frac{n!}{i! (n-1)!} [/itex]
     
  7. Sep 17, 2012 #6

    Ray Vickson

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    Of course the sum is still involved. The final answer must be in terms of n alone: it cannot contain "i", since all values of i have been summed over. Anyway, just multiplying and dividing by n! does not magically get rid of the sum.

    RGV
     
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