Binomial Theorem Class 12 Mathematics | Complete Note | PDF | Concept and Numericals

 In mathematics, especially in algebra and combinatorics, the binomial theorem is a fundamental idea. In Class 12 Math, students learn about the binomial theorem, which is used to expand statements of the type (a + b)ⁿ, where a and b are constants and n is a positive integer. Without having to manually multiply out each term in the extended expression, the theorem offers a rapid and effective method for determining the coefficients of the terms. The expansion formula, characteristics, and uses of the binomial theorem will all be covered in detail in this article. Students studying mathematics in Class 12 will have a better knowledge of the binomial theorem and how it can be applied in many mathematical circumstances after reading this article.

Binomial Theorem Class 12 Mathematics

Welcome to our article on the Binomial Theorem, a key section of mathematics for class 12 students. This article offers you a comprehensive answer that addresses all pertinent issues and includes examples, PDFs, theorems, and significant questions. We understand the importance it is to give you the resources you require to do well in your studies. You can learn the ideas by using our clear explanations, applicable examples, and crucial questions. So, look no farther than our blog post if you're searching for a comprehensive answer to the Binomial Theorem.

Explanation of Binomial Theorem

A binomial expression of the form (a + b)ⁿ, where n is a positive integer and a and b are constants, can be expanded using the binomial theorem formula. The theorem offers an alternative to manually multiplying out each term to get the coefficients of the terms in the extended expression.

For example expansion of ${\left(a+b\right)}^2$ is ${\left(a+b\right)}^2=a^2+2ab+b^2$

The expansion's terms have coefficients of 1, 2, and 1, respectively. The binomial coefficients, which are determined using Pascal's triangle, can be used to find these coefficients.

The binomial theorem can be extended to higher powers, such as (a + b)³, (a + b)⁴, and so forth, in addition to just squaring binomials. The general formula for (a + b)ⁿ expansion is:

${\left(a+b\right)}^n =C(n,0).a^{n-0}.b^0$$+ C\left(n,1\right).a^{n-1}.b^1 +$$C\left(n,2\right).a^{n-2}.b^2$$+ C\left(n,3\right).a^{n-3}.b^3$$+....+C\left(n,n-1\right).a^{n-(n-1)}.b^{n-1}$$+C\left(n,n\right).a^{n-n}.b^n$

The binomial coefficient, denoted C(n, r), is the combination of n things taken r at a time. This formula is a crucial tool in algebra and combinatorics since it helps us to find the coefficients of the terms in the expansion of any binomial expression.

Important of Binomial Theorem In Mathematics

 1. Simplifies calculations: Calculations are made easier by the binomial theorem, which offers a rapid and effective method of expanding binomial formulas without having to individually multiply out each term. Calculations may be done much more quickly and easily as a result.

2. Combinatorics: The binomial coefficients employed in the binomial theorem have combinatorial interpretations, which means they may be used to count the various ways an object can be selected from a collection. Because of this, the binomial theorem is a crucial combinatorics tool.

3. Probability: In probability theory, the binomial theorem is used to determine the probability that specific incidents will occur. The binomial theorem, for instance, can be used to determine the probability of getting a specific number of heads out of a specified number of coin flips.

4. Mathematical statistics: To model and interpret data, mathematical statistics also employs the binomial theorem. It is employed to figure out the probability that specific incidents will take place in a sample space.

5.Generalization: The multinomial theorem and the binomial series, which have applications in numerous branches of mathematics and science, can be extended from the binomial theorem to other mathematical ideas.

In conclusion, the binomial theorem is an important mathematical topic with applications in algebra, combinatorics, probability theory, mathematical statistics, and other areas.

Definition of Binomial Theorem 

Binomial coefficients, written as C(n, r), show how many possible methods there are to select r particular things from a set of n different objects. In the binomial theorem, the coefficients of the terms in the expansion of a binomial expression are determined using binomial coefficients.

The formula for the binomial theorem is :

$C(n,r) = \frac{n!}{r!.(n-r)!}$

where n! represents the factorial of n, or the product of all positive integers from 1 to n.

For example, C(6, 2) represents the number of ways to choose 2 objects from a set of 6 objects. Using the formula above, we can calculate C(6, 2) as:

$\mathrm{C}(6,2)=\frac{6!}{2!.\left(6-2\right)!}$$=\frac{1.2.3.4.5.6}{2!.4!}$$=\frac{\cancel{1}.\cancel{2}.\cancel{3}.\cancel{4}.5.6}{\cancel{1}.\cancel{2}.1.2.\cancel{3}.\cancel{4}}=\frac{30}{2}$$=15$

Therefore, there are 10 ways to choose 2 objects from a set of 5 objects.

Binomial coefficients have many applications in mathematics and science, particularly in combinatorics, probability theory, and statistics.

Pascal's Triangle And It's Relation To Binomial Coefficients

The triangle-shaped array of integers known as Pascal's triangle was created by French mathematician Blaise Pascal. Starting with a row carrying the number 1, the triangle is built by adding the neighboring numbers in the previous row to each row that follows. Here is a picture of the Pascal's triangle:

      1

     1 1

    1 2 1

   1 3 3 1

  1 4 6 4 1

An associated binomial coefficient may be found for each integer in Pascal's triangle. The coefficients of the terms in the expansion of (a + b)ⁿ are represented in the nth row of Pascal's triangle. For instance, the coefficients of the terms in the expansion of (a + b)⁴ correspond to the fourth row of Pascal's triangle:

      1
     4 1
    6 4 1
   4 6 4 1
  1 4 6 4 1

The binomial coefficient C(n, r) is found in the nth row and rth position of Pascal's triangle. For example, C(4, 2) is found in the 4th row and 2nd position of Pascal's triangle, which is 6.

For greater values of n, Pascal's triangle offers a straightforward method for calculating binomial coefficients. The triangle's symmetry and diagonals, among other intriguing characteristics and patterns, are revealed during the creation of Pascal's triangle.

In conclusion, the Pascal's triangle is a triangular array of numbers that represents the coefficients of a binomial. The triangle exposes a number of intriguing characteristics and patterns in addition to offering a practical method for computing binomial coefficients.

Binomial Theorem For Positive Integral Index

A binomial expression raised to a positive integral power can be expanded using the binomial theorem formula. An algebraic expression with two terms, as (a + b), is known as a binomial expression. The power to which the binomial expression is raised is referred to as the positive integral index, and it must to be a positive whole number.

 The formula for expanding a binomial expression raised to a power that is a positive whole integer is given by the binomial theorem for a positive integral index. Binomial coefficients are used in the formula to obtain the coefficients of the terms in the expansion. A useful technique in algebra, combinatorics, and probability theory is the binomial theorem.

General Term in Binomial Expansion

The general term in a binomial expansion refers to the term that represents the kth power of the first term (x) and the (n-r)th power of the second term (a), where n is the power to which the binomial expression is raised and k is a non-negative integer that represents the position of the term in the expansion.

Middle Terms in a Binomial Expansion

In a binomial expansion, the middle terms is the terms that are equal distant from the beginning and the end of the expansion. Specifically, the number of terms in the expansion must be odd, and the middle terms are the (n/2)th term and the ((n/2)+1)th term, where n is the power to which the binomial expression is raised.

Binomial Coefficient

A binomial coefficient, written as "C(n, r)", represents the number of ways to choose k objects from a set of n distinct objects, without regard to their order. It is also commonly known as a combination. 

C₀,C₁,C₂,C₃,C_4.....C_r....C_n-1  are called binomial coefficients.

Properties of Binomial Coefficients

There are some important properties of binomial coefficients  are : 

1. Symmetry Property: The binomial coefficient C(n, k) has the symmetry property of being equal to C(n, n-k). The fact that selecting k items from a collection of n items is equivalent to selecting n-k items from the same set gives birth to this characteristic.

2. Pascal's Identity : Pascal's Identity: According to Pascal's identity, C(n, k) + C(n, k-1) = C(n+1, k), where n and k are numbers in the range 0 to k, where n is a positive integer. This property results from the observation that, in order to select k items from a set of n objects, we can either select k-1 more objects from the remaining n-1 objects by including a specific object, or we can select k objects from the remaining n-1 objects by excluding that object.

3. Summation Property: The sum of binomial coefficients for a fixed value of n is equal to 2^n. That is,

C(n, 0) + C(n, 1) + C(n, 2) + ... + C(n, n) = 2^n.

This property arises from the fact that to form a subset of n objects, we have two choices for each object: either include it in the subset or exclude it from the subset. Therefore, the total number of subsets of a set of n objects is 2^n.

First Propertiy 

Second Property

Worked Out Example Of Binomial Theorem |PDF| 

This is  Worked example taken from Mathematics Fundamentals book . 

 Exercise of Binomial Theorem |PDF| 

This PDF offers a thorough reference for using the Binomial Theorem to solve problems. To assist you improve your understanding of the theorem, it offers solutions to issues and gives instances of exercises. You can deepen your comprehension of the Binomial Theorem and its uses in algebra by working through the provided activities. The tasks are made to test you and aid in the development of your problem-solving abilities while also reinforcing previously learnt ideas. This book will give you the confidence to approach and solve binomial issues, making it a great tool for both students and professionals alike.

 Exercise of Binomial Theorem |PDF| Exercise 2 

Next PDF is also for better practice without going anywhere. This pdf include concept with note and questions with answer , 

Credit : This PDF is taken from NepalEnotes.com blog website. You can also go for further information and related topics like study materials. 

Mind Map of Binomial Theorem Super Quick Revision 

Using this video you can revise all the topics and concept in short time . This video is in Hindi . Hope you will understand the concept of binomial theorem. 

Thank You for Watching .🙏

 Binomial Theorem for any Index 

We have discussed binomial theorem for positive integral index. In this section we discuss a more general binomial theorem in which the index or exponent is not necessarily a natural number. So, we state the theorem as given below. 

Theorem : Let n be a ratinal number and x be real number such that |x| < 1, then,

${\left(1+x\right)}^4=1+nx$$+\frac{n(n-1)}{2!}{x}^2$$+\frac{n(n-1)(n-2)}{3!}{x}^{3+....+}$$\frac{n\left(n-1\right)\left(n-2\right)....(n-r+1)}{r!}{x}^r$$+...$

The solution of this theorem is in the pdf. This pdf contains important deductions, different between the binomial theorem for positive integer exponent and for general exponent, workout example, and

Exponential and Logarithm Series

In this topic ,Euler's Number , Exponential series e. and containing five theorem with proof. with note and worked out examples, 

Source  : This pdf content is taken from Mathematics Fundamental Book . 
Editor : Prof. Kanhaiya Jha,Phd
Authors : Ramesh  Prashad Awasthi and Pradeep Bahadur Thapa 

Conclusion: 

A mathematical formula called the binomial theorem shows how to multiply a binomial equation by a positive integer power. This theorem heavily relies on the symmetry, Pascal's identity, summation property, and Vandermonde's identity of binomial coefficients, among other features. Due to these characteristics, the binomial theorem is a useful tool for algebraic expression simplification and for resolving issues in combinatorics, probability theory, and other branches of mathematics. Be sure to follow Edubook Ram Kumar Sah to discover further study resources on mathematics, including the binomial theorem and other ideas. Share this article at 3 student . 

Thank you for your important time to read this article 

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