Binomial Theorem

(for JEE, only positve integral indices).

Expansion
(a+b)n = nC0 anb0 + nC1 an-1b1+.....+ nCn a0bn

kthTerm
For (a+b)nTk = Tr+1

Tr+1= nCr an-rbr

Number of Terms
Number of Terms = n + 1

e.g. (a+b)89 has 90 terms in its expansion.

Middle Term
For (a+b)n.If n is even ; Middle Term = (n + 2) / 2

If n is odd Middle Terms = (n + 1) / 2 And (n + 3) / 2

Co-efficient of xn
For (ax + bx)n

Substitute in the kth term formula. Find r according to the given condition. Now apply the entire formula.

For constant term, x0

Binomial Theorem for any Index
For (a+b)n

nC0 = 1 ; nC1 = n ; nC2= n(n-1)(n-2) / 2!

... nCr = n(n-1(n-2).....(n-r) / r!

Negative Index
(Excluded from JEE)

(1+x)-1 = 1 - x + x2 - ..... + xn

(1 - x)-1= 1 + x + x2 + ...+ xn

(1 + x)-2= 1 - 2x +3x2 - 4x3 .....+ (n+1)xn

(1 - x)-2 = 1 + 2x + 3x2 + 4x3 +....+ (n+1)xn

Properties of Binomial Co-efficients
nC0+ .... + nCn = 2n

nC0 + nC2 + nC4+ ... + nCn= 2n-1