Binomial Theorem – Notes, Topics, Formulas, Equations, Books, Question & Answers

Binomial Theorem and its simple applications - Notes, Formula, Examples, Questions

Edited By Team Careers360 | Updated on May 07, 2022 01:41 PM IST

A binomial is an algebraic expression with two dissimilar terms connected by + or – sign. Let’s look into the following example to understand the difference between monomial, binomial and trinomial.

(Monomial term)
$x -y , \ y + 4$ (Binomial term)
$x^2 + y + 1$ (Trinomial term)

Note: Binomial Theorem is different than binomial distribution.

Binomial Theorem is a quick way of expanding a binomial expression with (that are raised to) large powers. This theorem is a really important topic(section) in algebra and has application in Permutations and Combinations, Probability, Matrices, and Mathematical Induction. If you are preparing for competitive exams for university admission or for jobs then this theorem is really important for you as it is a basic and important section of algebra. In this chapter, you will learn a shortcut that will allow you to find (x + y)ⁿ without multiplying the binomial by itself n times.

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Special cases of the binomial theorem were known since at least the 4th century BC when Greek Mathematician Euclid mentioned the special case of the binomial theorem for exponent 2. The binomial theorem for cubes was known by the 6th century in India. Isaac Newton is generally credited with the generalized binomial theorem, valid for any rational exponent.

Real-world use of Binomial Theorem:

The binomial theorem is used heavily in Statistical and Probability Analyses. It is so much useful as our economy depends on Statistical and Probability Analyses.
In higher mathematics and calculation, the Binomial Theorem is used in finding roots of equations in higher powers. Also, it is used in proving many important equations in physics and mathematics.
In Weather Forecast Services,
Ranking up candidates
Architecture, estimating cost in engineering projects.

Notes of Binomial Theorem

Important Topics of Binomial Theorem

Binomial Theorem for Positive Integral Index
Pascal's Triangle
General Term
Middle Term
Properties and Application of Binomial Theorem

Overview of Binomial Theorem

First, look into some following identities that you have done earlier

$(x+y)^0=1$

$(x+y)^1=x+y$

$(x+y)^2=(x+y)(x+y)=x^2+2xy+y^2$

$(x+y)^3=(x+y)(x^2+2xy+y^2)=x^3+3x^2y+3xy^2+y^3$

$(x+y)^3=(x+y)(x^3+3x^2y+3xy^2+y^3)=y^4+4x^3y+6x^2y^2+4xy^3+y^4$

Can you guess the next expansion for the binomial (x + y)⁵?

Now, you will observe that from above:

The total number of terms in the expansion is one more than the index. For example, in the expansion of (x + y)³ , a number of terms is 4 whereas the index of (x + y)² is 3.
The powers on x begin with n and decrease to 0 whereas the powers of the second quantity ‘y’ begin with 0 and increase to n.
In each term of the expansion, the sum of the indices of x and y is the same and is equal to the index of x + y.
The coefficients are symmetric.

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These patterns lead us to the Binomial Theorem, which can be used to expand any binomial.

The coefficients of the terms in the extension are the binomial coefficients.

Pascal's Triangle

If we arrange the coefficients in these expansions, this will look like

The structure given in the above figure looks like a triangle with 1 at the top vertex and running down the two slanting sides. The numbers in this array are known as Pascal’s triangle, after the name of French mathematician Blaise Pascal. It is also known as Meru Prastara by Pingla.

The General Term of Binomial Theorem

Expanding a binomial with a high exponent such as (x + 2y)¹⁶ can be a lengthy process. Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term. Note the pattern of coefficients in the expansion of (x + y)⁵

$(x+y)^{5}=x^{5}+\left(\begin{array}{c}{5} \\ {1}\end{array}\right) x^{4} y+\left(\begin{array}{c}{5} \\ {2}\end{array}\right) x^{3} y^{2}+\left(\begin{array}{c}{5} \\ {3}\end{array}\right) x^{2} y^{3}+\left(\begin{array}{c}{5} \\ {4}\end{array}\right) x y^{4}+y^{5}$

The second term is $\\\mathrm{\begin{pmatrix} 5\\1 \end{pmatrix}x^4y}$ . The third term is $\\\mathrm{\begin{pmatrix} 5\\2 \end{pmatrix}x^3y^2}$ . In this way, we can generalize this result.

The (r + 1)th term of the binomial expansion of (x + y)ⁿ is: $\\\mathrm{\begin{pmatrix} n\\r \end{pmatrix}x^{n-r}y^r}$ .

Middle term of Binomial Theorem

The middle term in the expansion (a + b)ⁿ , depends on the value of 'n'.

When 'n' is even

In this case, the number of terms in the expansion will be n + 1. Since n is, even so, n + 1 is odd. Therefore, the middle term is $\mathrm{\left ( \frac{n}{2}+1 \right )^{th}}$ term.

It is given by $\mathrm{T_{\frac{n}{2}+1}=\binom{n}{\frac{n}{2}}x^{\frac{n}{2}}y^{\frac{n}{2}}}$ .

When 'n' is odd

In this case, the number of terms in the expansion will be n + 1. Since n is, odd so, n + 1 is even. Therefore, there will be two middle terms in the expansion, namely $\mathrm{\left ( \frac{n+1}{2} \right )^{th}}$ and $\mathrm{\left ( \frac{n+3}{2} \right )^{th}}$ term.

And it is given by $\mathrm{T_{\frac{n+1}{2}}=\binom{n}{\frac{n-1}{2}}x^{\frac{n+1}{2}}\cdot y^{\frac{n-1}{2}}}$ and $\mathrm{T_{\frac{n+3}{2}}=\binom{n}{\frac{n+1}{2}}x^{\frac{n-1}{2}}\cdot y^{\frac{n+1}{2}}}$ .

Some Interesting Properties of Binomial Theorem:

The total number of each and every term in the expansion $( x + y )^n$ is n + 1 .
The sum total of the indices of x and y in each term is n .
The expansion shown above is also true when both x and y are complex numbers.
The coefficient of all the terms is equidistant (equal in distance from each other) from the beginning to the end.
The values of these binomial coefficients gradually go up to the maximum and progressively lessen.

Application of Binomial Theorem in finding e (Euler's Number):

e = 2.71828182846.......(digit goes on without repeating)

$e = (1 + {1}/{n})^n$

Let's use Binomial Theorem:

$(1 + {1}/{n})^n = \sum_{k=0}^{n}\binom{n}{k}1^{n-k}(\frac{1}{n})^k$ $=\sum_{k=0}^{n}\binom{n}{k}(\frac{1}{n})^k$

now for a most accurate value of e, n should be as big as possible ( $\lim_{n\rightarrow \infty}$ is representing that n should be the biggest possible number)

$=\lim_{n\rightarrow \infty}\sum_{k=0}^{n}\frac{n!}{k!(n-k)!} \cdot \frac{1}{n^k}$

Now,

$\lim_{n\rightarrow \infty}\frac{n!}{(n-k)!} \cdot \frac{1}{n^k} =\lim_{n\rightarrow \infty} \frac{n}{n}\cdot \frac{n-1}{n}\cdot \frac{n-2}{n} \cdot...... \cdot \frac{n-k+1}{n}$

$= 1\cdot 1\cdot 1\cdot ..... \cdot 1 \ \ (as \ n\rightarrow \infty)$

So the equation remains,

$\sum_{k=0}^{n}\frac{1}{k!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} +...... = 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + .....$

With calculating just a few terms we get e = 2.7083

The last expression is calculated using supercomputers to get the most accurate value of e.

Application of Binomial Theorem in Divisibility and Reminder Problems

if 7 divides 32³⁰, then the remainder is

we know that 32 = 2⁵, so, 32³⁰ can be written as

(2⁵)³⁰ = 2¹⁵⁰ = (2³)⁵⁰ = 8⁵⁰ = (7 + 1)⁵⁰

= [ (7)⁵⁰ + ⁵⁰C₁(7)⁴⁹ + ⁵⁰C₂ (7)⁴⁸ + ... + 1 ]

= [ 7( (7)⁴⁹ + ⁵⁰C₁(7)⁴⁸ + ⁵⁰C₂ (7)⁴⁷ + ... ) + 1 ]

= 7k + 1

⇒ remainder is 1.

Application of Binomial Theorem in finding the Greatest Term

Two important theorem

$\\\mathrm{\bullet \;\;2\leq\left ( 1+\frac{1}{n} \right )<3,\;\;n\geq1\;\;and\;n\in\mathbb{N}}\\\mathrm{\bullet \;\;If\;n>6,\;then\left ( \frac{n}{3} \right )^n<n!<\left ( \frac{n}{2} \right )^n}$

The above theorem can be used in solving problems such as, which one is greatest among 100¹⁰⁰ and (300)! ?

From the above results, $\mathrm{\left ( \frac{n}{3} \right )^n<n!}$

Put n = 300

= (100)³⁰⁰ < (300)! .........(i)

But, = (100)³⁰⁰ > (100)¹⁰⁰ .........(ii)

From (i) and (ii)

= (100)¹⁰⁰ < (100)³⁰⁰ < (300)!

Hence, (100)¹⁰⁰ < (300)!

How To Study Binomial Theorem :

First, you need to understand Binomial Theorem. You should be able to find the general term, greatest term, of any binomial equation. After getting a strong understanding of the theorem, you can jump on the application part.

If you are preparing for competitive exams then solve as many problems as you can. Do not jump on the solution right away. Remember if your basics are clear you should be able to solve any question on this topic.

NCERT Notes Subject wise link:

Important Books for Binomial Theorem:

Start from NCERT book, the illustration is simple and lucid. You should be able to understand most of the things. Solve all problems (including miscellaneous problem) of NCERT. If you do this, your basic level of preparation will be completed.

Then you can refer to the book Cengage Mathematics Algebra. The binomial theorem is explained very well in this book and there are lots of questions with crystal clear concepts. You can also refer to the book Arihant Algebra by SK Goyal or RD Sharma. But again the choice of reference book depends on person to person, find the book that best suits you the best depending on how well you are clear with the concepts and the difficulty of the questions you require.

NCERT Solutions Subject wise link:

NCERT Exemplar Solutions Subject wise link: