# Combinatorics Class - Sum Question

## Homework Statement

For any positive integer n determine:

$\sum\limits^n_{i=0} \frac{1}{i!(n-i)!}$

## 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

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jbunniii
Homework Helper
Gold Member
Hint: does this look familiar?

$$\frac{n!}{i!(n-i)!}$$

So the answer I'm looking for is

$\frac{\dbinom{n}{i}}{n!}$

Correct?

Or will it be

$\sum\limits^n_{i=0} \dfrac{\dbinom{n}{i}}{n!}$

I'm confused as to whether the sum is still involved.

you should find the following sum:

$\frac{1}{n!}*\sum \frac{n!}{i! (n-1)!}$

Ray Vickson
Homework Helper
Dearly Missed
Or will it be

$\sum\limits^n_{i=0} \dfrac{\dbinom{n}{i}}{n!}$

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