Binomial Theorem

The binomial theorem is one of the most popular and widely used theorems in mathematics. Generally, it is used for answering issues regarding disciples such as combinatorics, algebra, calculus, and probability. In probability, the major function of the binomial theorem is to understand whether the conducted experiment will succeed or fail. This theorem is used for operating on different types of number series like 1,2,3,4,5 and so on up to n. Today, we are going to discuss binomial theorems in detail. 

This article talks about binomial theorems. You will find brief information on the concept of the binomial theorem in maths, a thorough explanation of binomial coefficients, properties of binomial coefficients, and so on. So, let’s start by describing binomial theorems in the Maths study material. 

What is a binomial theorem? 

A binomial expression is described as (x+y) because it has two different variables x and y. When we consider (x+y)1, the resultant is (x+y). For (x+y)2, the expansion can be written as x2+2xy+y2. Getting answers to these equations is still easy but not when you have an expression of (x+y)n where n can be anything like an integer, a decimal, or a fraction. 

Mathematically, “n” belongs to the set of natural numbers and two independent variables such as a and b, which belong to the set of the primary number. The general formula for the binomial expansion is written as follows:

For negative variables, the binomial theorem can be applied. But the indices cannot be negative because, as per the expansion formula, the index value will indicate the upper limit of the sum of the series that cannot be negative. 

What are binomial coefficients? 

The coefficients in the binomial theorem are often termed as the integral values which are closely attached with the algebraic variables. Imagine the third element of an expansion series is 3C1a. Now, this term can be considered as the combined value that will be evaluated as:

3C1a = a x [3! / (3-1)!1!]

            = (3.2.1.a)/(2!)

            = 6a/2

            = 3a

In every binomial expansion, there are many terms involved and connected by the (+) operator, which are considered coefficients. Such as –

Similarly, if we consider the series of even and odd coefficients, the binomial theorem expressions can be written in the following manner:

C0 + C2 + C4 + C6 + C8 + C10 + C12 + C14 + C16 + ………. + C2n = 2n-1

C1 + C3 + C5 + C7 + C9 + C11 + C13 + C15 + C17 + ………. + C2n+1 = 2n

Properties of binomial coefficients

1- In case the binomial coefficients are added together, the sum will be 2n.

2- In case the binomial coefficients are placed with alternative addition and subtraction signs, the value will be 0.

3- For a binomial expansion series like nC1+2nC2+3nC3…..+nnCn, the resultant value will be n.2n-1.

4- If the binomial coefficient squares are added together, the total sum is given by [(2n)! / (n!)2].

5- The sum of even and odd terms in a binomial series will be the same as 2n-1.

Apart from these, there are some other properties of the binomial theorem. These are as follows:

Terms used in the binomial coefficients 

1. The general term is mostly denoted by using an alphabetical expression, as its value can change with the coefficient, its place in the series, and even the type of binomial expansion itself. It is often denoted by Ti+1.
2. The most significant term can be written in a complex formula where the mod operator is also used.
3. A term not having a particular variable, say x, is the independent term. So, for a binomial expression like [axp+(b/xq)]n, the independent term of x will be given as: 

4. Imagine two consecutive terms of a binomial expression: XR and xr+1. The coefficient of xr is nCr-1, and that of xr+1 is nCr. Then the ratio of these two consecutive terms or coefficients will be written as:

Conclusion 

A set of numbers can be operated easily by adding, multiplying, and so on. However, when dealing with the ‘n’ number of elements, finding the results can be difficult due to the lack of knowledge about the values. Therefore, some statistical analysis needs to be done. Before suggesting the binomial theorem, multiple ways were suggested such as mathematical and statistical calculations; however, the binomial theorem has proven to deliver the most accurate and precise results. 

In this article describing the binomial theorem, we studied the concept of the binomial theorem in length. We covered several other topics, such as binomial coefficients, properties of binomial coefficients, and other related topics. We hope this study material has helped you better understand the binomial theorem.