## 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*