Binomial Coefficients


by ritwik06
Tags: binomial, coefficients
ritwik06
ritwik06 is offline
#1
Aug18-08, 02:42 PM
P: 586
1. The problem statement, all variables and given/known data
If [tex]\sum^{n}_{r=0} \frac{1}{^{n}C_{r}} = a[/tex], then find the value of [tex]\sum^{n}_{r=0} \frac{r}{^{n}C_{r}}[/tex] 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.
Phys.Org News Partner Science news on Phys.org
Going nuts? Turkey looks to pistachios to heat new eco-city
Space-tested fluid flow concept advances infectious disease diagnoses
SpaceX launches supplies to space station (Update)
tiny-tim
tiny-tim is offline
#2
Aug18-08, 06:38 PM
Sci Advisor
HW Helper
Thanks
tiny-tim's Avatar
P: 26,167
Quote Quote by ritwik06 View Post
1. The problem statement, all variables and given/known data
If [tex]\sum^{n}_{r=0} \frac{1}{^{n}C_{r}} = a[/tex], then find the value of [tex]\sum^{n}_{r=0} \frac{r}{^{n}C_{r}}[/tex] in terms of a and n.[/tex]
Hi ritwik06!

Hint: suppose n = 12.

Then [tex]\sum^{n}_{r=0} \frac{1}{^{n}C_{r}}[/tex]

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

So what is [tex]\sum^{n}_{r=0} \frac{r}{^{n}C_{r}}[/tex] ?
Defennder
Defennder is offline
#3
Aug18-08, 09:52 PM
HW Helper
P: 2,618
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?

ritwik06
ritwik06 is offline
#4
Aug19-08, 12:59 PM
P: 586

Binomial Coefficients


Quote Quote by tiny-tim View Post
Hi ritwik06!

Hint: suppose n = 12.

Then [tex]\sum^{n}_{r=0} \frac{1}{^{n}C_{r}}[/tex]

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

So what is [tex]\sum^{n}_{r=0} \frac{r}{^{n}C_{r}}[/tex] ?
Thank god! Somebody helped me. But Tim, I wonder what you wish to convey... Please could you be more explicit
Dick
Dick is offline
#5
Aug19-08, 02:59 PM
Sci Advisor
HW Helper
Thanks
P: 25,168
Consider:
[tex]
\sum^{n}_{r=0} \frac{n-r}{^{n}C_{r}}
[/tex]
How does that compare with:
[tex]
\sum^{n}_{r=0} \frac{r}{^{n}C_{r}}
[/tex]
Does that give you any ideas??
tiny-tim
tiny-tim is offline
#6
Aug22-08, 03:44 PM
Sci Advisor
HW Helper
Thanks
tiny-tim's Avatar
P: 26,167
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!

(same for the other thread)
Dick
Dick is offline
#7
Aug23-08, 11:51 PM
Sci Advisor
HW Helper
Thanks
P: 25,168
That's nice of you to say, tiny-tim. Thanks. :) Now you've got me curious. ritwik06, did you get it? It's surprising easy if you think about it right, and pretty nonobvious if you don't. It took me a while.


Register to reply

Related Discussions
Binomial (Properties of Coefficients) Calculus & Beyond Homework 6
How is the negative binomial the inverse of the binomial distribution? Set Theory, Logic, Probability, Statistics 1
Binomial Coefficients Introductory Physics Homework 4
Binomial coefficients and pascal's triangle Introductory Physics Homework 2
Coefficients of friction Introductory Physics Homework 3