# Binomial Coefficients

## Homework Statement

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]

## 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.

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tiny-tim
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## Homework Statement

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]
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 $$\sum^{n}_{r=0} \frac{r}{^{n}C_{r}}$$ ? Defennder
Homework Helper
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?

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 $$\sum^{n}_{r=0} \frac{r}{^{n}C_{r}}$$ ? Thank god! Somebody helped me. But Tim, I wonder what you wish to convey... Please could you be more explicit Dick
Homework Helper
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??

tiny-tim
Homework Helper
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! 