Combinatorics Class - Sum Question

[theRukus]

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

 

[jbunniii]

Hint: does this look familiar?

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

 

[theRukus]

So the answer I'm looking for is

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

Correct?

 

[theRukus]

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.

 

[damabo]

you should find the following sum:

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

 

[Ray Vickson]

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