Binomial Theorem?

Question

Use the binomial theorem to prove:
http://img172.imageshack.us/img172/7686/ilc89pr2.jpg

Prove: http://img505.imageshack.us/img505/8370/ilc90blk3.jpg, 0 ≤ j ≤ n

Answer 1

Taking the second one first, because it's easy:
(n j) = n! / (j! (n-j)!) by definition
and similarly
(n n-j) = n! / ((n-j)! (n - (n-j))!)
= n! / ((n-j)! j!) since n - (n-j) = j
= (n j).

For the first one, simply expand (1-3)^n using the binomial theorem to get
(-2)^n = (1-3)^n
= (n 0) 1^n (-3)^0 + (n 1) 1^(n-1) (-3)^1 + (n 2) 1^(n-2) (-3)^2 + ... + (n n) 1^0 (-3)^n
= (n 0) (1) + (n 1) (-3) + (n 2) (9) + ... + (n n) (-3)^n
= (n 0) - 3 (n 1) + 9 (n 2) + ... + (n n) (-3)^n.