Combinatorics Class - Sum Question

  • Thread starter theRukus
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  • #1
theRukus
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Homework Statement


For any positive integer n determine:

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


Homework 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!!


The Attempt at a Solution

 

Answers and Replies

  • #2
jbunniii
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Hint: does this look familiar?

[tex]\frac{n!}{i!(n-i)!}[/tex]
 
  • #3
theRukus
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So the answer I'm looking for is

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

Correct?
 
  • #4
theRukus
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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.
 
  • #5
damabo
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you should find the following sum:

[itex] \frac{1}{n!}*\sum \frac{n!}{i! (n-1)!} [/itex]
 
  • #6
Ray Vickson
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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.

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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