(1+x)^{n} =
1+ nx + + ..... +

- Expansion is valid only when –1 < x < 1
^{n}C_{r}can not be used because it is defined only for natural number, so^{n}C_{r}will be written as- As the series never terminates, the number of terms in the series is infinite. .
- General term of the series (1+x)
^{-n}= T_{r+1}= (-1)^{r}x^{r} - General term of the series (1-x)
^{-n}= T_{r+1}= x^{r} - If first term is not 1, then make first term unity in the following way:
- (a+ x)
^{n}= a^{n}(1+x/a)^{n}if < 1

- (1+ x)
^{-1}= 1- x +x^{2}–x^{3}+ . . . + (-1)^{r}x^{r}+. . . - (1 - x)
^{-1}= 1+ x +x^{2}+x^{3}+ . . .+ x^{r }+ . . . - (1+ x)
^{-2}= 1- 2x +3x^{2}–4x^{3}+ . . .+ (-1)^{r}(r+1)x^{r}+. . . - (1 - x)
^{-2}= 1+ 2x +3x^{2}+4x^{3}+ . . .+ (r+1)x^{r}+. . . . - (1+x)
^{-3}= 1- 3x +6x^{2}–10x^{3}+. . .+ (-1)^{r }x^{r}+. . . - (1-x)
^{-3}= 1+ 3x +6x^{2}+10x^{3}+ . . .+ x^{r}+. . .

In the expansion of
(x_{1}+x_{2} + . . . + x_{n})^{m} where m, n ∈ N and x_{1}, x_{2}
, . . ., x_{n} are independent variables, we have .

- Total number of
term in the expansion =
^{m+n-1}C_{n-1} - Coefficient of ( where r
_{1}+ r_{2}+. . . .+ r_{n}= m, r_{i}∈Nis . - Sum of all the
coefficient is obtained by putting all the variables x
_{i }equal to1and is n^{m}.

Find the total number of terms in the expansion of (1 +
a + b)^{10} and coefficient of a^{2}b^{3}. .

Total number of
terms = ^{10+3 –1}C_{3-1} = ^{12}C_{2} = 66

Coefficient of a^{2}b^{3}
= = 2520

Find the number of
terms and coefficient of x^{5} in ( 1+x+x^{2})^{7}.

Here the variables
1, x, x^{2} are not independent so the general formula is not
applicable but as power of x varries from 0 to 14, therefore total number of
terms = 15. .

Now x^{5}
can be formed in three ways: 1^{2} x^{5}(x^{2})^{0},
1^{3} x^{3}( x^{2} )^{1} or

1^{0} x^{1}
(x^{2})^{2} , so total arrangement of coefficient

= =266

Alternative method:

coefficient of x^{5}
in ( 1+ x+x^{2})^{7}

= coefficient ox x^{5}
in ( 1- x^{3})^{7}( 1 - x)^{-7}

= coefficient of x^{5}
in ( 1 – 7x^{3}+ . . )(1 + ^{7}C_{1} x + ^{8}C_{2}
x^{2} + ^{9}C_{3} x^{3 }+ . . ¥ ).

= 462 – 28 ´ 7 = 462 – 196 = 266.