Binomial Theorem and Its Simple Applications

In mathematics, the binomial theorem is the most widely used theorem. It is used to answer issues in combinatorics, algebra, calculus, and probability, among other disciplines. It is used to compare two huge numbers, to discover the remainder when a value raised to a large exponent is divided by another number, and in probability to determine whether an experiment will succeed or fail. The binomial theorem is also used in weather forecasting, forecasting the national economy in the coming years, and IP address distribution.

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 the actual value of all these known expansions is 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, if we consider n, which belongs to the natural number set, and two independent variables like a and b, which belong to the primary number set, the general formula for the binomial expansion will be: = 88√3 It is possible to apply a binomial theorem for negative variables. 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, which can’t be negative.

What are Binomial Coefficients?

The coefficients in the binomial theorem can be defined as the integral value attached with the algebraic variables. For example, let’s say that 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. For example:
C₀ + C₁ + C₂ +C₃ +C₄ +C₅ +C₆ +C₇ +C₈ + ………. + C_n = 2ⁿ
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 + ………. + C_2n = 2^n-1
C₁ + C₃ + C₅ + C₇ + C₉ + C₁₁ + C₁₃ + C₁₅ + C₁₇ + ………. + C_2n+1 = 2ⁿ

Properties of Binomial Coefficients

1. When the binomial coefficients are added together, their total sum is 2ⁿ.
2. If the binomial coefficients are arranged with alternate plus and minus operators, the resultant value of the series will be 0.
3. When the squares of the binomial coefficients are added together, the total sum is given by [(2n)! / (n!)²].
4. The sum of even and odd terms in a binomial series will be the same as 2^n-1.
5. For a binomial expansion series like nC₁+2nC₂+3nC₃…..+nnC_n, the resultant value will be n.2^n-1.

Some 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)

1. Pascal’s rule:

(n+1/k) = (nk) + (n/k−1)

Terms Used in the Binomial Expansion

They are as follows:

• The general term is always 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.

• When a series like (a+b)ⁿ is considered, it has a middle term often used in calculations like arithmetic mean, geometric mean, statistics, and probability. The usual formula for the middle term depends on the value of the index. If n is even, the middle term will be noted as (n/2 +1). While for odd values, the [(n+1)/2]^th and [(n+3)/2]^{the term} that can be considered as the middle one.
• The most significant term can be written in a complex formula where the mod operator is also used.
• A term not having a particular variable, say x, is the independent term. So, for a binomial expression like [ax^p+(b/x^q)]ⁿ, the independent term of x will be given as:

T^r+1 = ⁿC_ra,^n-r.b^r where r = [np/(p+q)]

• Let’s consider two consecutive terms of a binomial expression: XR and x^r+1. The coefficient of x^r is nC_r-1, and that of x^r+1 is nC_r. Therefore, the ratio of these two consecutive terms or coefficients will be:

(nC_r/nC_r-1) = (n-r+1)/r

In What Ways Can Binomial Theorems be Used?

The binomial theorem has extensive use in mathematics, the instrumental field, and science. In the following section, a few such applications will be discussed for a better idea of its use.

1. It will become possible for anyone to find a remainder of numeric expressions like 12¹²/9 or 5⁵⁵⁶/4⁷⁸⁹. One can write these expressions in the form of binomial and apply the theorem properties to get the correct answer.
2. When you have to find a value of xⁿ, where x and n have values above 20, using the binomial theorem will be the best option. It will quickly give you the shortest possible answer with maximum accuracy and precision.
3. To solve mathematics problems like “Show that 12⁹ + 9¹² can be divided by 15”, the only option is to use the binomial expression.
4. AP, GP, and HP are derived from this binomial theorem only because they too consider a series of infinite lengths where the last number is represented by n.

Apart from these practical uses, the binomial theorem is also the basis for the concept of mathematical induction, advanced algebra, logarithm, permutation and combination, and even statistics.

Conclusion

The binomial theorem can be used to calculate a binomial statement that has been raised to a very big power. In Statistical and Probability Analysis, the binomial theorem is frequently employed. It is quite beneficial, as our economy is heavily reliant on statistical and probability analyses. The binomial theorem is used in advanced mathematics and calculating to determine roots of equations in higher powers. This theorem has applications in Permutations and Combinations, Probability, Matrices, and Mathematical Induction, and is a very important part of algebra.