1. Binomial Theorem for any + ve integral index:
(x+a)n = nC0 xn + nC1 xn-1 + nC2 xn-1 C2 xn-2 a2 + ……. + nCr xn-r ar + ….. + nCn an
(i) General term – Tr+1 = nCr xn-r is the (r+1) term form beginning.
(ii) (m+1)th term from the end = (n-m+1)th from beginning = Tn=m+1
(iii) middle term
(a) If n is even then middle term = {(n/2) +1}th term
(b) If n is odd then middle term = {(n+1)/2}th and {(n+3)/2}th term
Binomial coefficient of middle of middle term is the greatest binomial coefficient.
2. To determine a particular term in the given expasion:
Let the given expansion be {xa ± (1/xb)}n , if xn occurs in Tr+1 (r+1)th term then r is given by n a - r (a + b) = m and for x0, n a - r (a + b) = 0
3. Properties of Binomial coefficients:
For the sake of convenience the coefficient nC0, nC1, nC2 ….. nCr …… nCn are usually denoted by C0, C1 , …….. Cr ……. Cn respectively.
* C0 + C1 + C2 + ….. + Cn = 2n
* C0 - C1 + C2 - C3 + …… + Cn = 0
* C0 + C2 + C4 + …… = C1 +C3 + C5 + …… = 2n-1
* nCr = (n/r) n-1Cr-1 = (n/r ) (n-1) / (r-1) n-2Cr-2 and so on …….
* 
* nCr + nCr-1 = n+1Cr
* C1 + 2C2 + 3C3 + ….. + nCn = n.2n-1
* C1 - 2C2 + 3C3 ……. = 0
* C0 + 2C1 + 3C2 + ….. + (n+1) Cn = (n+2)2n-1
* C20 + C21 + C22 + ….. + C2n = {(2n)! / (v!)2} = 2nCn
* C20 - C21 + C22 - C2n = {(2n)! / (n!)2 } = 2nCn
* C20 - C21 + C22 - C23 + …….
Note : 2n+1C0 + 2n+1C0 + 2n+1C1 + …… + 2n+1Cn = 2n+1Cn+1 + 2n+1C n+2 + ……. 2n+1C2n+1 = 22n
* C0 + (C1 /2) + (C2 3) + ……. + {Cn / (n+1) } = {(2n+1 -1) / (n+1)}
* C0 - (C1/2) + (C2/3) – (C3 /4) ….. + {(-1)nCn / (n+1) } = (1/(n+1))
4. Greatest term:
(i) If {(n+1)a / (x+a) } Î Z (integer) then the expansion has two greatest term. These are kth and (k+1)th where x & a are + ve real nos.
(ii) If {(n+1)a / (x+a)} Î Z then the expansion has only one greatest term . this is (k+1)th term k = [(n+1)a / (x+a) ], {[.] denotes greatest integer less then or equal to x }
5. Multinomial Theorem:
(i) (x+a)n = r=0ån nCr xn-r ar, n Î N
(ii) (x+y+z)n = å(r+s+t=n) {(n!)/(s!r!t!)} xr yszt
Generalized (x1+x2+ ….. xk)n
6. Total no. of terms in the expansion (x1+x2 + ….. xn)m is m+n-1Cn-1
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