# Binomial Coefficients

1. Aug 18, 2008

### ritwik06

1. The problem statement, all variables and given/known data
If $$\sum^{n}_{r=0} \frac{1}{^{n}C_{r}} = a$$, then find the value of $$\sum^{n}_{r=0} \frac{r}{^{n}C_{r}}$$ in terms of a and n.[/tex]

3. The attempt at a solution
I tried to write down the terms of both the series, but to no avail. i cant think of anything.Please shed some light.

2. Aug 18, 2008

### tiny-tim

Hi ritwik06!

Hint: suppose n = 12.

Then $$\sum^{n}_{r=0} \frac{1}{^{n}C_{r}}$$

= (0!12! + 1!11! + 2!10! + 3!9! + …)/12!

So what is $$\sum^{n}_{r=0} \frac{r}{^{n}C_{r}}$$ ?

3. Aug 18, 2008

### Defennder

Hi tim, I'm not seeing how this helps to solve the problem. You have a term dependent r in each summand, so how do we express it in a?

4. Aug 19, 2008

### ritwik06

Thank god! Somebody helped me. But Tim, I wonder what you wish to convey... Please could you be more explicit

5. Aug 19, 2008

### Dick

Consider:
$$\sum^{n}_{r=0} \frac{n-r}{^{n}C_{r}}$$
How does that compare with:
$$\sum^{n}_{r=0} \frac{r}{^{n}C_{r}}$$
Does that give you any ideas??

6. Aug 22, 2008

### tiny-tim

Hi ritwik06!!

Have you got this now … you haven't said?

If you haven't, then follow Dick's hint … it's much better than mine!