The Binomial Theorem is an essential tool for simplifying long expressions that follow the pattern (a + b)n. By providing a concise formula for expanding these expressions, it makes calculations more efficient. Mastery of this theorem streamlines calculations and facilitates diverse applications across mathematics, from combinatorics to probability theory.
The Binomial Theorem is a mathematical theorem that provides a formula for expanding the powers of binomials. It states that for any real numbers a and b and any non-negative integer n, the expansion of (a+b)n is given by:
Where nCk represents the binomial coefficient, and equals
The Binomial Theorem refers to the mathematical principle that allows for the expansion of any positive integral power of a binomial expression into a series format. (This theorem was given by Newton)
Pascal’s Triangle is a visual representation used to find binomial coefficients. It provides a convenient method for expressing the expansion of the coefficients of binomial expansions
The binomial theorem is a fundamental result in algebra that provides a formula for expanding expressions of the form (a + b)n, where "a" and "b" are any numbers or variables, and "n" is a non-negative integer. The binomial theorem equation is typically written as:
This equation expresses the expansion of the binomial expression (a +b)n into a sum of terms, where nCr represents the binomial coefficient "n choose r."
(a + b)n = nC0 an + nC1 an-1 b + nC2an–2b2 + … + nCr an-r br + …+ nCn-1 a1bn–1 + nCn bn
=
Some more important formulas of binomial theorem are:
(x +a)n = nC0 xn + nC1 xn-1 a1 + nC2 xn–2 a2 + … + nCn an
(x – a)n = nC0 xn – nC1 xn-1 a1 + nC2 xn–2 a2 – … + nCn (–a)n
Property – 1 nCr × r! = nPr
Property – 2 nCr = nCn–r
if nCx = nCy ⇒ x = y or x + y = n
Property – 3
Property – 4 nCr + nCr+1 = n+1Cr+1
Property – 5
Property – 6
Using the binomial theorem expansion formula, we get:
= 3C0 (2x)3 (–3y)0 + 3C1 (2x)2 (–3y)1 + 3C2 (2x)1 (–3y)2 + 3C3 (2x)0 (–3y)3
= 8x3 – 36x2y + 54xy2 – 27y3
Using the binomial theorem expansion formula, we get:
(x +2)4 = 4C0 (x)4 + 4C1 (x)3 (2)1 + 4C2 (x)2 (2)2 + 4C3 (x)1 (2)3 + 4C4 (x)0 (2)4 .
=1.x4.1 + 4.x3.2 + 6.x2.4 + 4.x.8 + 1.16
=x4 + 8x3 + 24x2 + 32x + 16
In binomial expansion, the general term and the middle term are usually asked to be found.
General Term of a Binomial Expansion (a +b)n is expressed as:
Tr+1 = nCr an–r br.
In the binomial expansion (a+b)n, there are n+1 terms.
The independent term in the binomial expansion refers to the term that does not contain any variables, i.e., it is the constant term.
Independent term is obtained by writing a general term and equating the power of the variable to 0.
Lets understand this concept with the help of an example:
Find the term independent of x in the expansion of ?
Given :
Let (r + 1)th term be the independent term for the given binomial expansion.
We know that the general term is Tr+1 = nCr . an–r . br
Here, n = 9, a = 3x2 /2 and b = (–1/3x)
T(r+1) = 9Cr. (3x2/2)9 - r . (–1/3x)r
⇒ T(r+1) = (–1)r. 9Cr . (3)9-2r. (2)r-9. (x)18-3r …….. (1)
∵ (r + 1)th term is an independent term
⇒ x18-3r = x0
⇒ 18 -3r = 0 ⇒ r = 6
By substituting r = 6 in equation (1) we get,
⇒ T(6+1) = 9C6 . (6)-3 = 7/18
Solution: We know that the general term is Tr+1 = nCr an–r br.
= T(r+1) = 3Cr x3–r. (2)r.
Solution: Here n is even, the middle term is
(2 + 1)th term = 3rd term
T3 = T2+1 = 4C2 (2y)2 (–3)2
6.4.y2.9 = 216y2
Solution: Tr+1 = nCr xn–r yr
Tr+1 = (–1)r 5Cr . (2)5–r . x 10–2r . x–3r .3–r
= x10–2r–3r = x0
10–5r = 0 ⇒ 10 = 5r ⇒ r = 2
Substituting r = 2
Solution: Given equation can be rewritten as (x2 + 2x + 1)4
Now, (x2 +2x+1)4 = [(x+1)2]4 = (x + 1)8
This implies, number of terms are n + 1 = 9
Solution: (x + y + z)n can be expanded as-
(x + y + z)n ={(x + y) + z}n
= nC0 (x + y)n + nC1 (x + y)n–1.z + nC2(x + y)n–2.z2 + … + nCnzn
=(n + 1) terms + n terms + (n – 1) terms + … + 1 term
Total number of terms = (n +1) + n + (n – 1) + … + 1
=(n +1)(n +2)/2
a.
b.
c.
d.
Solution: The correct answer is C :
(n + 1)th term from the end in the expansion of (2x − 1/x )3n is T2n + 1 from the beginning.
Ans: - The binomial theorem is a mathematical theorem that provides a formula for expanding the powers of binomials. It states that for any real numbers a and b and any non-negative integer n, the expansion of (a+b)n is given by:
(Session 2025 - 26)