JEE Main Important Chapter - Binomial Theorem and Its Simple Applications

The expansion of the equation will become longer as the power rises and it will be very mind-numbing to calculate. Here comes the solution; a binomial expression has been improved to solve a very large power with ease by using the binomial theorem. 

Let’s study all the facts associated with binomial theorem such as its definition, properties, examples, applications, etc. It will clarify all your doubts regarding the binomial theorem.

We can explain a binomial theorem as the technique to expand an expression which has been elevated to any finite power. It is a powerful tool for the expansion of the equation which has a vast use in Algebra, probability, etc.

JEE Main Maths Chapters 2024 

Binomial Theorem Expansion 

In binomial theorem expansion, the binomial expression is most important in an algebraic equation which holds two different terms. 

Such as: a + b, a3 + b3, etc.

Let’s consider; x, y ∈ R; n ∈ N 

Then the result will be 

\[\sum_{i=0}^{n}nC_rx^{n-r}.y^r + nC_rx^{n-r}.y^r + ...nC_{n-1}x.y^{n-1} + nC_n.y^n\]

i.e. \[(x+y)^n\] = \[\sum_{i=0}^{n}nC_rx^{n-r}.y^r\]

Where \[nC_r\] = \[\frac{n!}{(n-r)!r!}\]

Binomial Theorem Examples

Example 1: 

Expand the given equation (x/2 + 3/y)4.

Sol: Using the equation, \[(x+y)^n\] = \[\sum_{i=0}^{n}nC_rx^{n-r}.y^r\] , we get: 

\[\frac{x}{2} + \frac{3}{y})^4 = 4C_0(\frac{x}{2})^4 + 4C_1(\frac{x}{2})^3(\frac{3}{y}) + 4C_2(\frac{x}{2})^2(\frac{3}{y})^2+ 4C_3(\frac{x}{2})(\frac{3}{y})^3 + 4C_4(\frac{3}{y})^4\]

Example 2: Expand the given equation (√3 + 1)5 + (√3 − 1)5

Sol:

We have

\[(x+y)^5+(x-y)^5\] = \[2(5C_0x^5 + 5C_2x^3y^2 + 5C_4xy^4)\]

= \[2(x^5 + 10x^3y^2 + 5xy^4)\]

After applying it  to the equation now,

=  \[(\sqrt{3}+ 1)^5 - (\sqrt{3}- 1)^5\]

= 2\[(\sqrt{3}^5)\] + 10\[(\sqrt{3}^3)(1)^2\] +5\[(\sqrt{3})(1)^4\]

= \[88 \sqrt{3}\] 

How Binomial Expansion Formula is Asked in JEE Main Exam?

In JEE Main exams, questions related to the binomial expansion formula are commonly framed to test candidates' understanding of algebraic manipulations and the application of mathematical concepts. Here's how the binomial expansion formula might be presented in JEE Main questions:

1. Finding a Specific Term:

Candidates may be asked to find a specific term in the expansion of a binomial expression raised to a certain power.

For example:

$\text{Find the coefficient of } x^3 \text{ in the expansion of } (2x - 3)^5$ .

2. Identifying Patterns:

Questions may involve recognizing patterns or relationships within the binomial expansion. For instance:

$\text{If } (a + b)^n \text{ has the general term } T_k = 84a^{n-k}b^k, \text{ find the values of } n \text{ and } k$ .

3. Manipulating Expressions:

Candidates might be required to simplify or manipulate binomial expressions.

For example: $\text{Simplify the expression } (x + 2y)^4 - (x - 2y)^4 \text{ using the binomial expansion.}$

4. Application in Probability:

Binomial expansion is often related to probability problems. Questions may involve calculating probabilities using the binomial distribution. For instance:

$\text{In a binomial distribution with } n = 8 \text{ trials and success probability } p = 0.3, \text{ find the probability of exactly 3 successes.}$

5. Combining with Other Concepts:

Questions may integrate the binomial expansion with other mathematical concepts, such as sequences and series. For example:

$\text{If the expansion of } (a - b)^n \text{ is } a^n - \binom{n}{1}a^{n-1}b + \ldots, \text{ find the sum of the coefficients in the expansion.}$

Binomial Coefficients

In the binomial theorem, the coefficients are the integral values associated with the algebraic variables.

There are many terms involved in every binomial expansion that are connected by the (+) operator and are referred to as coefficients. As an example:

$C_0 + C_1 + C_2 +C_3 +C_4 +C_5 +C_6 +C_7 +C_8 +.............. + C_n = 2^n$

Similarly, given a series of even and odd coefficients, the binomial theorem expressions can be written as follows:

$C_0 + C_2 + C_4 + C_6 + C_8 + C_{10} + C_{12} + C_{14} + C_{16} +.................... + C_{2n} = 2^{n-1}$

$C_1 + C_3 + C_5 + C_7 + C_9 + C_{11} + C_{13} + C_{15} + C_{17} +.................... + C^{2n+1} = 2^n$

Significant Facts for Summing up Binomial Expansion

1. The whole amount of terms in the expansion of (x + y)n are (n + 1).

2. The summation of exponents of x and y is always n.

3. Binomial coefficients are known as nC0, nC1, nC2,…up to n Cn, and similarly signified by C0, C1, C2, ….., Cn.

4. The binomial coefficients which are intermediate from the start and the finish are equal i.e. nC0 = nCn, nC1 = nCn-1, nC2 = nCn-2,….. etc.

Also, we can apply Pascal’s triangle to find binomial coefficients.

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Also, there are some other important expansions that you must learn about. They are given below and many useful binomial expansions.

1. (1+x)n + (1 − x)n = 2[K0 + K2 x2 + K4 x4 + …]

2. (x + y)n – (x−y)n = 2[K1 xn-1 y + K3 xn-3 y3 + K5 xn-5 y5 + …]

3. (1 + x)n = nΣr-0 n Cr . x r = [K0 + K1 x + K2 x2 + … Kn xn]

4. (x + y)n + (x−y)n = 2[K0 xn + K2 xn-1 y2 + K4 xn-4 y4 + …]

5. (1+x)n − (1−x)n = 2[K1 x + K3 x3 + K5 x5 + …]

6. In the expansion (x + a)n − (x−a)n ; the number of terms of are (n/2) if “n” is even or (n+1)/2 if “n” is odd.

7. In the expansion (x + a)n + (x−a)n; the number of terms are (n+2)/2 if “n” is even or (n+1)/2 if “n” is odd.

Binomial Coefficient Properties

Here, I am considering the binomial coefficient as K. Some of the most important properties of binomial coefficients are:

1. K0 + K2 + K4 + … = K1 + K3 + K5 + … = 2n-1

2. K0 + K1 + K2 + … + Kn = 2n

3. nK1 + 2.nK2 + 3.nK3 + … + n.nKn = n.2n-1

4. K0 – K1 + K2 – K3 + … +(−1)n.nKn = 0

5. K02 + K12 + K22 + …Kn2 = [(2n)!/ (n!)2]

6. K1 – 2K2 + 3K3 – 4K4 + … +(−1)n-1 Kn = 0 for n > 1

Other Properties of Binomial Coefficients

1. Symmetry property: (n/x) = (n/(n−x))

2. Special cases: (n/0) = (n/n) = 1

3. Negated upper index of binomial coefficient:

for k≥0

(n/k) = (−1)k((k−n−1/k)

4. Pascal’s rule:

(n+1/k) = (nk) + (n/k−1)

Binomial Theorem and Pascal’s Triangle:

Pascal’s triangle is a triangular pattern of numbers formulated by Blaise Pascal. The binomial expansion of terms can be represented using Pascal's triangle. To understand how to do it, let us take an example of a binomial (a + b) which is raised to the power ‘n’ and let ‘n’ be any whole number. For assigning the values of ‘n’ as {0, 1, 2 …..}, the binomial expansions of (a+b)n for different values of ‘n’ as shown below.

With this kind of representation, the following observations are to be made.

• Each expansion has one term more than the chosen value of ‘n’.

• In each term of the expansion, the sum of the powers is equal to the initial value of ‘n’ chosen.

• The powers of ‘a’ start with the chosen value of ‘n’ and decreases to zero across the terms in expansion whereas the powers of ‘b’ start with zero and attains value of ‘n’ which is the maximum.

• The coefficients start with 1, increase till half way and decrease by the same amounts to end with one. 

Important Points to Remember While Solving Binomial Expansion:

• The total number of terms in the expansion of (x + y)\[^{n}\] is (n+1)

• The sum of exponents is always equal to n i.e (x + y) = n.

• $^nC_{0}, ^nC_{1}, ^nC_{2}, … .., ^nC_{n}$ is called binomial coefficients and also represented by C\[_{0}\], CC\[_{1}\], CC\[_{2}\] ….., C\[_{n}\] respectively.

• The binomial coefficients which are equidistant from the beginning and from the ending are of equal value i.e. $^nC_0= ^nC_n,^nC_1= ^nC_{n-1} , n^C_2= n^C_{n-2} $,….. etc.

• To find binomial coefficients we can also use Pascal’s Triangle.

Some of The Terms Used in Binomial Expansion

1. General Term: The general term in binomial expansion is represented using letters because its value can change based on different factors like the coefficient, its position in the series, and the type of binomial expansion being considered.

2. Middle Term: When you have a series like (a+b)^n, it often has a middle term that's important in various calculations like finding averages, statistical analysis, and probability. The formula for the middle term depends on whether 'n' is even or odd. For even 'n', the middle term is at position (n/2 +1), and for odd 'n', there are two middle terms at positions [(n+1)/2] and [(n+3)/2].

3. Most Significant Term: The most significant term in a binomial expansion can be expressed using a complex formula that may involve the mod operator.

4. Independent Term: This term doesn't have a specific variable, like 'x', in it. So, in an expression like [ax^p + (b/x^q)]^n, the independent term involving 'x' can be found using the formula Tr+1 = nCra^(n-r)b^r, where r is calculated as [np/(p+q)].

5. Consecutive Terms Ratio: When you have two consecutive terms in a binomial expression, say XR and xr+1, the coefficient of xr is nCr-1, and the coefficient of xr+1 is nCr. To find the ratio of these two consecutive terms or coefficients, you can use the formula (nCr/nCr-1) = (n-r+1)/r.

Uses of Binomial Theorems

• The binomial theorem can be used to expand binomial expressions of any power. This is useful for simplifying expressions and solving equations.

• The binomial theorem can be used to approximate functions using polynomials. This is useful for estimating values and simplifying calculations.

• The binomial theorem can be used to calculate probabilities in binomial distributions. This is useful for modeling real-world phenomena such as coin flips and dice rolls.

• The binomial theorem can be used to analyze financial data such as stock prices and interest rates. This is useful for making investment decisions and managing risk.

Application of Binomial Theorem

1. To Calculate the value ‘e’ (Euler's Number)

As we know, e = 2.71828182846...

Here, e = (1 + 1/n)n

Now it is time to apply Binomial Theorem:

(1+1/n)n=\[\sum_{k=0}^{n}(nk)1^{(n-k)}(\frac{1}{n})^k\]= \[\sum_{k=0}^{n}(nk)(\frac{1}{n})^k\]

To obtain the most precise value of e, the amount of ‘n’ should be as large as possible.

Here, \[\lim_{n\rightarrow\infty}\] is expressing that ‘n’ should be the largest possible number.

= \[\lim_{n\rightarrow\infty}\sum_{K=0}^{n}\frac{n!}{k!(n-k)!}.\frac{1}{n^k}\]

Here,

\[\lim_{n\rightarrow\infty}\frac{n!}{k!(n-k)!}.\frac{1}{n^k}\] = \[\lim_{n\rightarrow\infty}\frac{n}{n}.\frac{n-1}{n}\frac{n-2}{n}...\frac{n-k}{n}\] = 1⋅1⋅1⋅……1(as n→∞) 

Now the obtained equation can be written as:

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

By calculating the above equation for certain terms we calculate that

e = 2.7083

The final expression is obtained by the use of supercomputers to find the most perfect value of e.

2. Divisibility and remainder problems is an application of binomial theorem 

Example- If 3230 is divided by 7, then what will be the remainder?

As we know, 25 = 32, 

3230 can thus be written as:

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

= [ (7)50 + 50K1(7)49 + 50K2 (7)48 + ...  + 1]

= [7 ( (7)49 + 50K1(7)48  +  50K2 (7)47  + ... ) + 1 ]

  Now, (7k + 1)/7, gives us the remainder 1

3. Concluding the highest term is an application of binomial theorem 

• If n>6, then \[(\frac{n}{3})^n\] < n! < \[(\frac{n}{2})^n\]

• n ≥ 1 and n ∈ N, 2 ≤ (1+1/n) < 3

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

From the above results,

Insert n = 300  

= (100)300 < (300)! .........(i)

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

From equation (i) and (ii)

= (100)100 < (100)300 < (300)!   

Therefore, (100)100 < (300)!

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Conclusion

In this article, we have elaborated on concepts of the Binomial Theorem and Its Simple Applications, a crucial topic for JEE Main. We will also learn the fundamental concepts and problem-solving strategies related to this theorem. Whether you're looking for a binomial theorem formula or insights into its properties, you will find it all right here in one place. We have provided comprehensive explanations, definitions, and practice questions in our downloadable PDFs which is FREE. Mastering this topic will greatly enhance your exam performance, so don't miss out on this valuable resource.