Binomial Expansion Question Driving Me Mad

[demon8991]

Binomial Expansion Question Driving Me Mad!

http://img489.imageshack.us/img489/5239/binomialiv6.jpg

This is the last question on my maths sheet, and i must have been staring at it for hours, I've read all my notes and book but i just can't piece it together at all.

Driving me crazy!

You guys got any pointers, I am really confused

 

[acm]

a) (1 + x)^n = (n,0) + (n,1)x + ... + (n,n)x^n
(1 + x)^2n = (1 + x)^n(1 + x)^n
Observe that x^n-2 * x^2 = x^n , the Hint shows that the coefficient (n,n-2) = (n,2) thus the coefficient of x^n will be (n,2)^2 and so forth for all powers of n. So it can be assumed that the coefficient of x^n = (n,0)^2 + (n,1)^2 + ... + (n,n)^2.

b) Obvious, by expansion. (1 + x)^2n = (2n, 0) + ... + (2n,n)x^n + ...+ (2n,2n)x^2n.

c)By equating coefficients (2n,n) = (n,0)^2 + (n,1)^2 + ... + (n,n)^2
The definition of (2n,n) = (2n)!/n!(2n-n)! = (2n)!/(n!)^2 as required.