Binomial Theorem: Mean, SD, Properties & Related Terms

Read further to learn about the binomial theorem, its formula, its expansion, and step by step explanation.

The binomial theorem is one of the most frequently used equations in the field of mathematics and also has a large number of applications in various other fields. Some of the real-world applications of the binomial theorem include:

  • The distribution of IP Addresses to the computers.
  • Prediction of various factors related to the economy of the nation.
  • Weather forecasting.
  • Architecture.

Binomial theorem, also sometimes known as the binomial expansion, is used in statistics, algebra, probability, and various other mathematics and physics fields. The binomial theorem is denoted by the formula below: 

(x+y)n =r=0nCrn. xn-r. yr

where, n N and x,y R

What is a Binomial Experiment?

The binomial theorem formula is generally used for calculating the probability of the outcome of a binomial experiment. A binomial experiment is an event that can have only two outcomes. For example, predicting rain on a particular day; the result can only be one of the two cases – either it will rain on that day, or it will not rain that day.

Since there are only two fixed outcomes to a situation, it’s referred to as a binomial experiment. You can find lots of examples of binomial experiments in your daily life. Tossing a coin, winning a race, etc. are binomial experiments. 

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What is a binomial distribution?

The binomial distribution can be termed as the measure of probability for something to happen or not happen in a binomial experiment. It  is generally represented as:

p: The probability that a particular outcome will happen

n: The number of times we perform the experiment

Here are some examples to help you understand, 

  • If we roll the dice 10 times, then n = 10 and p for 1,2,3,4,5 and 6 will be ⅙. 
  • If we toss a coin for 15 times, then n = 15 and p for heads and tails will be ½. 

There are a lot of terms related to the binomial distribution, which can help you find valuable insights about any problem. Let us look at the two main terms, standard deviation and mean of the binomial distribution. 

Standard Deviation of a Binomial Distribution

The standard deviation of a binomial distribution is determined by the formula below: 

= npq


n = Number of trials

p = The probability of successful trial

q = 1-p = The probability of a failed trial

Read: Binomial Coefficient

Mean of a Binomial Distribution

The mean of a binomial distribution is determined by, 

= n*p


n = Number of trials

p = The probability of successful trial

Introduction to the Binomial Theorem

The binomial theorem can be seen as a method to expand a finite power expression. There are a few things you need to keep in mind about a binomial expansion: 

  • For an equation (x+y)n the number of terms in this expansion is n+1.
  • In the binomial expansion, the sum of exponents of both terms is n.
  • C0n, C1n, C2n, …. is called the binomial coefficients.
  • The binomial coefficients which are at an equal distance from beginning and end are always equal.

Coefficients of all the terms can be found by looking at Pascal’s Triangle. 

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Binomial Theorem Explained | Statement

The process of extending an expression that was increased to any finite power uses the binomial theorem. A binomial theorem is a formidable expansionary technique with uses in probability, algebra, and other fields.

Statement: The binomial theorem states that any non-negative power of a binomial (x + y) may be expanded into a sum of the form, 

Binomial Expansion Formula: Let n ∈ N,x,y,∈ R then,

(x + y)n = nΣr=0 nCr xn – r · yr where,

With the binomial theorem explained, proceed to learn some of its applications. People use binomial theorem for various mathematical operations, such as determining the remainder of the digits of a number. 

Terms related to Binomial Theorem

Let us now look at the most frequently used terms with the binomial theorem

  • General Term

The general term in the binomial theorem can be referred to as a generic equation for any given term, which will correspond to that specific term if we insert the necessary values in that equation. It is usually represented as Tr+1.

Tr+1=Crn . xn-r . yr

  • Middle Term

The middle term of the binomial theorem can be referred to as the value of the middle term in the expansion of the binomial theorem. 

If the number of terms in the expansion is even, the (n/2 + 1)th term is the middle term, and if the number of terms in the binomial expansion is odd, then  [(n+1)/2]th and [(n+3)/2)th are the middle terms. 

  • Independent Term

The term which is independent of the variables in the expansion of an expression is called the independent term. The independent term in the expansion of axp + (b/xq)]n is

Tr+1 = nCr an-r br, where r = (np/p+q) , which is an integer.

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Properties of Binomial Theorem

  1. C0 + C1 + C2 + … + Cn = 2n
  2. C0 + C2 + C4 + … = C1 + C3 + C5 + … = 2n-1
  3. C0 – C1 + C2 – C3 + … +(−1)n . nCn = 0
  4. nC1 + 2.nC2 + 3.nC3 + … + n.nCn = n.2n-1
  5. C1 − 2C2 + 3C3 − 4C4 + … +(−1)n-1 Cn = 0 for n > 1
  6. C02 + C12 + C22 + …Cn2 = [(2n)!/ (n!)2]

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Binomial Expansion Terms

Finding the general term or the middle term is a common task in binomial expansion. Here, various words from the binomial expansion are discussed, including 

General term Middle term Independent term
Finding a particular term Numerically greatest term Ratio of coefficients

Binomial Theorem Formula:

Any power of a binomial can be expanded into a series using the binomial theorem formula. The binomial theorem formula is (a+b)n= ∑nr=0nCr an-rbrr, where r is a real number, a and b are real numbers, and n is a positive integer. 

The binomial formulas (x + a)10, (2x + 5)3, (x – (1/x))4, and so on can be expanded with the use of this formula. An expansion of a binomial raised to a specific power is made possible with the use of the binomial theorem formula. In the parts that follow, let’s examine the binomial theorem formula and its use. 

The binomial theorem states: If x and y are real numbers, then for all n ∈ N,

(x+y)n = nC0 xny0 + nC1 xn-1y1 + nC2 xn-2 y2 + … + nCk xn-kyk +….+ nCn x0yn

⇒ (x + y)n = ∑nk=0nCk xn-kyk

where, nCr = n! / [r! (n – r)!]

Binomial Theorem: Points To Remember!

The binomial theorem can be better understood if you remember the following important details. 

  • There are n + 1 terms in total in the binomial expansion of (x + y)n. 
  • The total of the x and y powers in each term of the expansion of (x + y)n equals n. 
  • Both sides of the expansion have equivalent binomial coefficient values. 
  • The binomial expansion of the expression (x + y + z)n has n(n + 1) terms. 

Binomial Theorem Formula: Application

The binomial expressions are expanded using the binomial theorem formula. Calculus, combinatorics, probability, and other crucial fields of mathematics are all subject to the formula’s application. 

By way of illustration, (101)5 = (100+1)5 = 1005+ 5 1004 + 10 1003+ 10 1002+ 5 100 + 1 = 10,000,000,000+ 500,000,000 + 10,000,000 + 100,000 + 500 +1 = 10,510,100,501.

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The binomial theorem is one of the most used formulas used in mathematics. It has one of the most important uses in statistics, which is used to solve problems in data science. 

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What is the use of the Binomial theorem in data science?

A Binomial Theorem is one in which just two outcomes are conceivable, such as success or failure, gain or loss, win or lose, and the chance of success and failure is the same for all trials. Each trial is independent since the last toss has no bearing on or affects the result of the current toss. A binomial experiment has just two possible outcomes and is repeated n times. A binomial distribution has two parameters: n and p, where n is the total number of trials and p is the probability of success in each trial. Discrete probability distributions are used in data science to model binary and multi-class classification problems and evaluate the performance of binary classification models, such as calculating confidence intervals, and model the distribution of words in the text for natural language processing.

Is binomial theorem tough?

Once students are familiar with the derivation, the notion of the binomial theorem becomes simple to grasp. The Binomial theorem describes how to extend statements of the type (a+b)^n, such as (x+y)^7. The greater the power, the more difficult it is to raise statements like this directly. The Binomial theorem, on the other hand, makes the operation pretty quick! The Binomial Theorem is a simple method for expanding a binomial equation with (that are raised to) high powers. This theorem is a crucial topic (part) in algebra, with applications in Permutations and Combinations, Probability, Matrices, and Mathematical Induction.

What are some real-life use cases of binomial theorems?

The binomial theorem is frequently employed in statistical and probability analyses. It is incredibly beneficial because our economy is based on statistical and probability calculations. The Binomial Theorem is used in advanced mathematics and computing to identify roots of equations in higher powers. It is also used in the proof of many significant equations in physics and mathematics. It has applications in Weather Forecast Services, Architecture Candidate Ranking, Cost Estimation in Engineering Projects, and the number of faulty items in a batch. The binomial is employed in real-life circumstances where dichotomies appear.

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