Binomial Theorem

Binomial Theorem

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.

1.0Binomial Theorem Statement

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:

$(a + b)^{n} = \sum_{k = 0}^{n}^{n} C_{k} a^{n - k} b^{k}$

$(a + b)^{n} = \sum_{k = 0}^{n} C_{n}^{k} x_{n}^{n - k} y^{k}$

Where ⁿC_k represents the binomial coefficient, and equals $\frac{n !}{k ! ( n - k )!}$

2.0Define Binomial Theorem

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

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

3.0Binomial Theorem Equation

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:

$(a + b)^{n} = \sum_{r = 0}^{n}^{n} C_{r} a^{n - r} b^{r}$

This equation expresses the expansion of the binomial expression (a +b)ⁿ into a sum of terms, where ⁿC_r represents the binomial coefficient "n choose r."

4.0Binomial Coefficients Expansion

(a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ a^n-1b + ⁿC₂a^n–2b² + … + ⁿC_r a^n-r b^r + …+ ⁿC_n-1 a¹b^n–1 + ⁿC_n bⁿ

= $\sum_{r = 0}^{n}^{n} C_{r} a^{n - r} b^{r}$

Some more important formulas of binomial theorem are:

(x +a)ⁿ = ⁿC₀xⁿ + ⁿC₁ x^n-1 a¹ + ⁿC₂ x^n–2 a² + … + ⁿC_n aⁿ

(x – a)ⁿ = ⁿC₀xⁿ – ⁿC₁ x^n-1 a¹ + ⁿC₂ x^n–2 a² – … + ⁿC_n (–a)ⁿ

5.0Binomial Theorem Properties

Property – 1 ⁿC_r × r! = ⁿP_r

Property – 2 ⁿC_r = ⁿC_n–r

if ⁿC_x = ⁿC_y ⇒ x = y or x + y = n

Property – 3 $\frac{^{n} C _{r}}{^{n} C _{r - 1}} = \frac{n - ( r - 1 )}{r}$

Property – 4 ⁿC_r + ⁿC_r+1 = ⁿ⁺¹C_r+1

Property – 5 $^{n} C_{r} = \frac{n}{r}^{n - 1} C_{r - 1}$

Property – 6 $\frac{^{n} C _{r}}{r + 1} = \frac{^{n + 1} C _{r + 1}}{n + 1}$

6.0Binomial Theorem Examples

Expand (2x–3y)³.

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

Expand x+24

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

7.0Binomial Expansion Terms

In binomial expansion, the general term and the middle term are usually asked to be found.

General Term
Middle Term
Independent Term

General Term of Binomial Expansion

General Term of a Binomial Expansion (a +b)ⁿ is expressed as:

T_r+1 = ⁿC_r a^n–r b^r.

In the binomial expansion (a+b)ⁿ, there are n+1 terms.

Middle Term of Binomial Expansion

When n is even, then the middle term will be = $(\frac{n}{2} + 1)^{th} t er m$
When n is odd, then there are two middle terms; they will be $\frac{( n + 1 ) ^{th}}{2} t er man d \frac{( n + 1 )}{2} + 1 t h t er m$

Independent Term of Binomial Expansion

The independent term in the binomial expansion refers to the term that does not contain any variables, i.e., it is the constant term.

$(a + b)^{n} = \sum_{r = 0}^{n}^{n} C_{r} a^{n - r} b^{r}$

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 $(\frac{3 x ^{2}}{2} - \frac{1}{3 x})^{9}$ ?

Given : $(\frac{3 x ^{2}}{2} - \frac{1}{3 x})^{9}$

Let (r + 1)th term be the independent term for the given binomial expansion.

We know that the general term is T_r+1 = ⁿC_r . a^n–r . b^r

Here, n = 9, a = 3x² /2 and b = (–1/3x)

T_(r+1) = ⁹C_r. (3x²/2)^{9 - r} . (–1/3x)^r

⇒ T_(r+1) = (–1)^r. ⁹C_r . (3)^9-2r. (2)^r-9. (x)^18-3r …….. (1)

∵ (r + 1)th term is an independent term

⇒ x^18-3r = x⁰

⇒ 18 -3r = 0 ⇒ r = 6

By substituting r = 6 in equation (1) we get,

⇒ T₍₆₊₁₎ = ⁹C₆ . (6)^-3 = 7/18

8.0Binomial Theorem Solved Questions

Write the general term of (x+2)³

Solution: We know that the general term is T_r+1 = ⁿC_r a^n–r b^r.

= T_(r+1) = ³C_r x^3–r. (2)^r.

Find the middle term in the expansion of (2y–3)⁴

Solution: Here n is even, the middle term is

(2 + 1)^th term = 3^rd term

T₃ = T₂₊₁ = ⁴C₂ (2y)² (–3)²

6.4.y².9 = 216y²

Find the constant term in the expansion of $(2 x^{2} - \frac{1}{3 x ^{3}})^{5}$

Solution: T_r+1 = ⁿC_r x^n–r y^r

$T_{r + 1} =^{5} C_{r} (2 x^{2})^{5 - r} (\frac{1}{3 x ^{3}})^{r}$

T_r+1 = (–1)^r ⁵C_r . (2)^5–r . x ^10–2r . x^–3r.3^–r

= x^10–2r–3r = x⁰

10–5r = 0 ⇒ 10 = 5r ⇒ r = 2

Substituting r = 2

$T_{2 + 1} =^{5} C_{2} (2)^{3} \frac{1}{3} = 10 \cdot \frac{8}{9} = \frac{80}{9}$

Find the number of terms in the expansion of (1+ 2x + x²)⁴

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

Find the number of terms in the expansion of (x + y + z)ⁿ

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 + … + ⁿC_nzⁿ

=(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

9.0Binomial Theorem JEE Mains Questions

The (n+1)th term from the end in the expansion of (2x − 1/x )³ⁿ is :

a. $\frac{2 n !}{3 n ! n !} \cdot 2^{n} \cdot x^{- n}$

b. $\frac{n !}{3 n ! 3 n !} \cdot 2^{n} \cdot x^{- n}$

c. $\frac{3 n !}{2 n ! n !} \cdot 2^{n} \cdot x^{- n}$

d. $\frac{3 n !}{2 n !} \cdot 2^{n} \cdot x^{- n}$

Solution: The correct answer is C :

$\frac{3 n !}{2 n ! n !} \cdot 2^{n} \cdot x^{- n}$

(n + 1)^th term from the end in the expansion of (2x − 1/x )³ⁿ is T_{2n + 1} from the beginning.

10.0Solved Question on Binomial Theorem

What is the binomial theorem?

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:

$(a + b)^{n} = \sum_{k = 0}^{n}^{n} C_{k} a^{n - k} b^{k}$

Frequently Asked Questions

The constant term in a binomial expansion is the term that does not contain any variables (e.g., x or y). It is obtained when all the variables in the expansion are raised to the power of zero.

The numerically greatest term in a binomial expansion is the term with the highest absolute value.

No, the binomial theorem specifically applies to binomial expressions, which consist of two terms raised to a power. However, similar techniques can be used to expand expressions with more than two terms, such as trinomials or multinomials.

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