Any algebraic expression consisting of only two terms is known as a binomial expression. It's expansion in power of x is known as the binomial expansion.

e.g. (i) a + x (ii) a^{2} + (iii) 4x – 6y

Such formula by which any power of a binomial expression can be expanded in the form of a series is known as binomial theorem. For a positive integer n , the expansion is given by .

(a+x)^{n}
= ^{n}C_{0}a^{n}+^{n}C_{1}a^{n–1}
x+^{n}C_{2} a^{n-2} x^{2} + . . . + ^{n}C_{r}^{
}a^{n–r} x^{r} + . . . + ^{n}C_{n}x^{n}.

where ^{n}C_{0} , ^{n}C_{1}
, ^{n}C_{2} , . . . , ^{n}C_{n} are called
Binomial co-efficients. The value of ^{n}C_{r} is defined as

Similarly (a – x)^{n} = ^{n}C_{0}a^{n}–^{n}C_{1}a^{n–1}
x+^{n}C_{2} a^{n-2} x^{2} – . . . +(–1)^{r}
^{n}C_{r}^{ }a^{n–r} x^{r}+ . . . +(–1)^{n
n}C_{n}x^{n}.

Expand

= ^{7}C_{0}x^{7}+^{7}C_{1}x^{6}+^{7}C_{2}x^{5}+^{7}C_{3}x^{4}
+ ^{7}C_{4}x^{3}

+^{7}C_{5}x^{2} +^{7}C_{6}
x+^{7}C_{7}

= x^{7} + 7x^{5}
+ 21x^{3} + 35 x + + ++ .

The general term in
the expansion of ( a+x)^{n} is (r+1)^{th} term given as t_{r+1}
= ^{n}C_{r} a^{n-r} x^{r} Similarly the
general term in the expansion of ( x + a)^{n} given as t_{r+1}
= ^{n}C_{r} x^{n-r} a^{r}. The terms are
considered from the beginning. .

The (r + 1)^{th}
term from the end = ( n – r + 1)^{th} term from the beginning . .

Corollary:

Coefficient of x^{r} in expansion
of (1 + x)^{n} is ^{n}C_{r} .

Find the
co-efficient of x^{24} in .

Since, general term ((r+1) th term) in

= ^{15}C_{r}(x^{2})^{15–r}

= ^{15}C_{r}
x^{30–2r }= ^{15}C_{r}^{
}3^{r}a^{r} x^{30–3r}

If this term
contains x^{24}. Then .

30–3r = 24

&⇒ 3r = 6

&⇒ r = 2

Therefore, the
co-efficient of x^{24} = ^{15}C_{2} ´9a^{2}.

If the co-efficient
of (2r+4)th term and (r–2) th term in the expansion of (1+x)^{18} are
equal, find r .

Since, co-efficient
of (2r+4) th term in (1+x)^{18} = ^{18}C_{2r+3}

Co-efficient of
(r–2) th term = ^{18}C_{r–3}

&⇒ ^{18}C_{2r+3 }= ^{18}C_{r–3}

&⇒ 2r+3+r–3 = 18

&⇒ 3r = 18

&⇒ r = 6

The binomial
coefficients in the expansion of (a+x)^{n} equidistant from the
beginning and the end are equal. .

· When n is even

Middle term of the expansion is term.

· When n is odd

In this case term and term are the middle terms.

e.g. Middle term in the expansion of (1
+ x)^{4} and (1 + x)^{5}.

Expansion of (1
+ x)^{4} has 5 terms, so third term is middle term i.e. term.

Expansion of (1 + x)^{5} has 6
terms, so 3rd and 4th both are middle terms, that is, th and th terms are middle
terms.

Find the middle
term in the expression of ( 1- 2x + x^{2})^{n}.

( 1- 2x + x^{2})^{n}
= [ ( 1- x)^{2}]^{n} = ( 1- x)^{2n}

Here 2n is even integer, therefore, th i.e. ( n+1) th term will be the middle term

Now (n+1) th term
in (1 - x)^{2n} = ^{2n}C_{n} (1)^{2n-n}(-x)^{n}

= ^{2n}C_{n}(-x)^{n}
=.

Prove that middle term in the expansion of is .

Since 2n is even, therefore th i.e. (n+1)th term will be the middle term.

Now (n+1) th term i.e. middle term in is given by

t_{n+1} = ^{2n}C_{n}x^{2n-n}=^{2n}C_{n}
x^{n}

=

=

= .

To determine the greatest coefficient in the
binomial expansion of (1 + x)^{n} , consider the following

=

Now
the (r + 1)^{th} binomial coefficient will be greater than the rth
binomial coefficient when,

= >1

&⇒ .......(1)

But r must be an integer, and therefore when
n is even, the greatest binomial coefficient is given by the greatest value of
r_{ }, consistent with (1) i.e., r = and hence the greatest
binomial coefficient is ^{n}C_{n/2}.

Similarly if n be odd, the greatest binomial coefficient is given when,

Show that the greatest coefficient in the expansion of is .

Since middle term has the greatest coefficient,

So, greatest coefficient = coefficient of middle term

= ^{2n}C_{n}
= .

To determine the
numerically greatest term (absolute value) in the expansion of (a + x)^{n},
when n is a positive integer. Consider.

Thus |T_{r + 1}|
> |T_{r}| if

. . . (2)

Note:

.ThusT_{r+1}
will be the greatest term if, r has the greatest value consistent with
inequality (2)

Find the greatest
term in the expansion of (2 + 3x)^{9} if x = 3/2.

Therefore T_{r+1}
≥ T_{r} if,

90 - 9r ≥ 4r

&⇒ 90 ≥
13r

, r being an integer, hence r = 6.

T_{r+1} = T_{7}
= T_{6+1} = ^{9}C_{6} (2)^{3} (3x)^{6 }

Find the greatest term in the expansion of .

Let rth term be the greatest term.

Since , now

&⇒
.
. . . (1)

Now again£ 1

&⇒ . . . (2)

From (1) and (2) follows that

&⇒ r = 8 is the greatest term and
its value

= = .

There are (n + 1)
terms in the expansion of (a+b)^{n}, the first and the last term being
a^{n} and b^{n} respectively.

Example.1 .

If in the expansion of ( 1+x)^{43},
the coefficient of (2r+1)th term is equal to the coefficient of (r+2)th term
find r.

Given in the
expansion of (1+ x)^{43}, the coefficient of (2r +1)th term = the
coefficient of (r +1)th term so, ^{43}C_{2r} = ^{43}C_{r+1}

&⇒ 2r + r + 1 = 43 or r = 14.

for the binomial expansion (a + x)^{n}.

The Pascal’s triangle is way of directly calculating binomial coefficients for different indices. It is given as follows.

How to construct a Pascal's triangle is explained below

The coefficients in the Binomial Expansion can be easily determined with the help of Pascal's triangle.

- D
^{m}(ax + b)^{n}= 0 if m > n where D^{m}_{ }is m^{th}derivative w. r. t. x . . - D
^{m}(ax + b)^{n}= a^{n}n! if n = m

D^{m} (ax + b)^{n}