1. Binomial Theorem
If n is a positive integer, then binomial theorem is
(x+y)n = nc0.xn + nc1xn-1y + nc2xn-2y2 + nc3.xn-3y3 + ……. + ncrxn-ryr + …. + ncn.yn
2. General Term in a binomial expansion:
In the binomial expansion of (x+y)n , general term is denoted by Tr + 1 and it is
Tr + 1 = ncr.xn – r.yr
3. Combinations or groups formula:
ncr = n!/[( n – r ) !].[r!]
4. Middle term in a binomial expansion:
In the binomial expansion of (x+y)n, middle term is T( n/2 + 1) if n is even, and T(n + 1)/2 and T( n + 3)/2 , if n is odd.
5. Binomial Coefficients in the binomial expansion (x+y)n
nC0, nC1, nC2, nC3,….. nCr… nCn are called Binomial Coefficients.
6. Binomial Coefficient of xm in (axp + b / xq )
The value of r of the term which contains the coefficient of xm is
(np – m )/( p + q)
7. Independent Term of x in (axp + b / xq )
The value of r of the term which does not contain x is
( np ) / (p + q)
8. Greatest Binomial Coefficients:
In the binomial expansion of (x + y)n , the greatest binomial coefficient is
nc(n+1)/2 , nc( n + 3 )/2 , when n is an odd integer, and nc( n/2 + 1) , when n is an even integer.
9. Numerically Greatest term in the binomial expansion: (1 + x)n
In the binomial expansion of (1 + x)n, the numerically greatest term is found by the following method:
If [( n + 1 ) | x | ] / [| x | + 1] = K + f,
Where K is an integer and f is a positive proper fraction, then
( K + 1) th term is the numerically greatest fraction.
And if [( n + 1 ) | x | ] / [| x | + 1] = K,
Where K is an integer, then
Kth term and (K + 1)th terms are the two numerically greatest terms.
10. In the binomial expansion of (x+y)n :
1. Sum of the binomial coefficients is 2n
nc0 + nc1 + nc2 + …………. + ncn = 2n
2. Sum of the odd binomial coefficients is 2n – 1
c1 + c3 + c5 + …………. = 2n – 1
3. Sum of the even binomial coefficients is 2n – 1
c0 + c2 + c4 +……….. = 2n – 1
11. Number of terms in various expansions:
Number of terms in the expansion of
1. ( x + y )n is n + 1
2. ( x + y + z ) n is [( n + 1 ) ( n + 2 )]/2
3. ( x + y + z + w) n = [ ( n + 1)(n + 2 ) ( n + 3 )]/ 1. 2.3