A Binomial Expression
Any algebraic expression consisting of only two terms is known as a binomial expression. It's expansion in powers of x is known as the binomial expansion.
For example:
(i) a + x
(ii) a2 +
(iii) 4x - 6y
Binomial Theorem:
Such formula by which any power of a binomial expression can be expanded in the form of a series is known as binomial theorem. When n is a positive integer,
(a + x)n = nC0an+nC1an-1x + nC2 an-2 x2 + ... + nCr an-r xr + ... + nCnxn,
where nC0, nC1, nC2 .... nCn are called Binomial co-efficients.
Properties of the Binomial Expansion
There are (n + 1) terms in the expansion of (a + b)n, the first and the last term being an and bn respectively. If nCx = nCy, then either x = y or x + y = n
nCr = nCn - r =
. e.g. 12C9 = 12C3.The general term (which is (r + 1)th term) is = nCr an - r br. It is usually denoted by Tr+1.
In the summation notation, the binomial theorem is written as
.n is called the index of the binomial. The values of the binomial coefficients are symmetrically placed about the middle value if the index n is even, and about two middle values (which
are equal) if the index is odd.
Middle term
(i) When n is even
Middle term of the expansion is the
term i.e., nCn/2 an/2 bn/2 in the expansion of (a + b)n.(ii) When n is odd
Middle terms of the expansion are the
term and the 
term.These are given by,
in the expansion of (a + b)n.Note:
1. The binomial co-efficients in the expansion of (a+x)n, equidistant from the beginning and from the end are equal.
2. The binomial coefficient of the rth term from the end = binomial coefficient of the (n - r +2)th term from the beginning.
3. If there are two middle terms, then the binomial co-efficients of two middle terms will be equal and those two co-efficients will be greatest.
Greatest Binomial Coefficient
To determine the greatest coefficient in the binomial expansion, (1 + x)n, when n is a positive integer.
Coefficient of
= 
Now the (r + 1)th binomial coefficient will be greater than the rth binomial coefficient when
Tr+1
Tr 
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 nCn/2.
Similarly if n be odd, the greatest binomial coefficient is given when,
.To determine the numerically greatest term in the expansion of (a + x)n, when n is a positive integer. Consider
.Thus |Tr + 1| > |Tr| if
. .... (2)Note:
.Thus Tr+1 will be the greatest term, if r has the greatest value as per relation (2).
Tips to Remember
(a)
for the binomial expansion of (a + x)n.(b) (n + 1)Cr = nCr + nCr-1
(c) When n is even,
(x + a)n + (x - a)n = 2(xn + nC2 xn-2 a2 + nC4 xn-4 a4 + .... +nCn an).
When n is odd,
(x + a)n + (x - a)n = 2(xn + nC2 xn-2 a2+ ... +nCn-1 x an-1).
When n is even,
(x + a)n - (x - a)n = 2(nC1 xn-1a + nC3 xn-3a3 + ... +nCn-1 x an-1).
When n is odd
(x + a)n - (x - a)n = 2(nC1xn-1a + nC3xn-3 a3 + ... +nCn an).
Properties of Binomial Coefficients
For the sake of convenience, the coefficients nCo, nC1,....nCr,.....nCn are usually denoted by Co, C1,... Cr ... Cn respectively.
Put x = 1 in (A) and get, 2n = Co + C1 +......+Cn
or Co + C1+ ..... + Cn = 2n. ....(D)
Also putting x = - 1 in (A) we get,
0 = Co - C1 + C2 - C3 +.......
C0 + C2 + C4 + ........= C1 + C3 + C5 +.... But from (D) Co + C2 + C4 + ....... + C1 + C3 + C5 +.......=2n.
Hence Co + C2 + C4 + ......... = C1 + C3 + C5 +.......= 2n-1.
For x = 2, (A) gives nC0 + 2 nC1 + 22 nC2 + ... + 2n nCn = 3n .
For Problem related to series of binomial coefficients in which each term is a product of two binomial coefficients.
If sum of lower suffices of binomial expansion in each term is the same
i.e. nC0 nCn + nC1 nCn-1 + nC2 nCn-2 + ....+ nCn nC0
i.e. 0 + n = 1 + (n - 1) = 2 + (n - 2) =....= n + 0.
Then the series represents the coefficient of xn in the multiplication of the following two series
(1 + x)n = C0 + C1x + C2x2 + . . + Cnxn
and (x + 1)n = C0xn + C1xn-1 + C2xn-2 + . . + Cn.
&font=(font)&format=(format)&size=(size)&color=(color)&font=britanic&format=png&color=hh0000&text=%C2%A9Copyright%20KAPIL%20ARYA%202010.&clientAction=wml:anchor.click&size=15){kind=small}
&font=(font)&format=(format)&size=(size)&color=(color)&font=britanic&format=png&color=bb0000&text=Download%20Solved%20Papers&clientAction=wml:anchor.click&size=15){kind=small}
&font=(font)&format=(format)&size=(size)&color=(color)&font=britanic&format=png&color=bb0000&text=Concept%20Bank&clientAction=wml:anchor.click&size=20){kind=small}