# Summation with binomial coefficients question

## Homework Statement

##\sum\limits_{r=0}^n\frac{1}{^nC_r}=a##. Then find the value of $$\sum\sum\limits_{0\le i<j\le n}(\frac{i}{^nC_i}+\frac{j}{^nC_j})$$

## Homework Equations

I have used two equations which I derived myself. This is the first one.

The second one is:

3. The Attempt at a Solution

Using first equation and second equation:

Now I have to subtract the cases where I=j to get the required sum. But Iis the above conclusion correct? Because I am not getting the required answer after subtracting.

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haruspex
Homework Helper
Gold Member
In the last line of the derivation of your second equation, you have been inconsistent in your substitutions of m for n. Two m's should be m+1.
But I don't understand how you use this equation anyway. a is a function of n, but where you use the equation you seem to be using it as a generic fact for any m, without changing a. I.e. you cannot now substitute n for m as being equal.

Then how do you find ##(2n+1)\sum\limits_{r=0}^n\frac{r}{^nC_r}##?

haruspex