## anonymous 4 years ago the coefficient of x^2 in the expansion of (1 + x/5)^n, where n is a positive integer, is 3/5. Find the value of n. How to do this???? :P

1. anonymous

$(1+x)^n=1+nx+\dbinom{n}{2}x^2+...$ in your case you will have $1+n\frac{x}{5}+\dbinom{n}{2}\frac{x^2}{25}+ ...$

2. anonymous

so basically i need to multiply 1/25 and $\left(\begin{matrix}n \\ 2\end{matrix}\right)$, but how do I do that?

3. anonymous

and equate it to 3/5.

4. anonymous

so you know that $\frac{n(n-1)}{2}\times \frac{1}{25}=\frac{3}{5}$ $\frac{n(n-1)}{2\times 25}=\frac{3}{5}$ $\frac{n(n-1)}{10}=3$ $n(n-1)=30$

5. anonymous

and since n is a whole number, only possible answer is $n=6$

6. anonymous

wait, sorry, I don't get why n(n - 1)/2

7. anonymous

oh ok lets go slower

8. anonymous

to compute $\dbinom{n}{k}$ you do not really use $\frac{n!}{k!(n-k)!}$ because it it is too much work, you just cancel. but we can write it out anyway

9. anonymous

$\dbinom{n}{2}=\frac{n!}{2!(n-2)!}$ ok?

10. anonymous

ohhh. so if it was the coefficient of x^3, then we would do n(n - 1)(n - 3)/3 ??

11. anonymous

then numerator is $n\times (n-1)\times (n-2)!$ so you can cancel the $(n-2)!$ and get $\frac{n(n-1)}{2}$

12. anonymous

it would be $\frac{n(n-1)(n-2)}{3!}$

13. anonymous

don't forget the factorial in the denominator

14. anonymous

^sorry, that's what I meant to type :P Got it.

15. anonymous

great!

16. anonymous

So, what's up, satellite??? haven't been here in a while :P

17. anonymous

not much. where you been?

18. anonymous

Sitting at home, studying for my mocks -_-. Ohhh, and i've got braces. i look WEIRD.

19. anonymous

been a month it seems

20. anonymous

ouch. did they hurt?

21. anonymous

well, i know. ^nopee, not much, but after a few hours they're tickling me :P And I feel like a kid, 'cause over here most people had their braces off by the 7th / 9th grade, and i'm starting in the 10th.

22. anonymous

But i've got a pretty blue, AND it's only for a year, so it's cool.

23. anonymous

well at least they are a color and not those awful railroad tracks!

24. anonymous

*7th/8th grade. I KNOW, those are horrid. My brother didn't put any colours - I killed him for it. Idiot.

25. anonymous

More than that, I know some people who have gotton the clear, translucent braces, not the metal ones, and they've put colours on them. Kinda kills the whole point of see-through ones, doesn't it???

26. anonymous

the new thing seems to be to make you have them twice. once around 6th /7th, then a year off, then back again. i think it is just a way to get more money out of parents

27. anonymous

What cheapsters.

28. anonymous

See??/ i'm such a nice girl, I became your fan :P