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Of course, you need a little math to describe a binomial series. That’s where a binomial theorem comes into action. With the increase the power of the factor, increases the expansion and thus that becomes lengthy and complicated to calculate. A binomial expression that has been raised to any infinite power can be easily calculated using the Binomial Theorem formula. The binomial expansions formulas are used to identify probabilities for binomial events (that have two options, like heads or tails). A binomial distribution is the probability of something happening in an event. The binomial theorem widely used in statistics is simply a formula as below :

\[(x+a)^n\] =\[ \sum_{k=0}^{n}(^n_k)x^ka^{n-k}\]

Where,

∑ = known as “Sigma Notation” used to sum all the terms in expansion frm k=0 to k=n

n = positive integer power of algebraic equation

\[(^n_k)\] = read as “n choose k”

Binomial theorem (sometimes called the binomial expansion) is a mathematical statement that expresses for any positive integer n, the nth power of the sum of two numbers a and b may be demonstrated as the sum of n + 1 terms of the form. Thus, the Binomial Theorem communicates that, where n is a positive integer:

(p + q) n = pn + (nC1) pn-1q + (nC2) pn-2q2 + … + (nCn-1) pqn-1 + qn

If you were to roll a dice 15 times, the probability of you rolling a 5 is 1 out of 6 1/6. This ends in a binomial distribution of (n = 15, p = 1/5).

Knowing the odds of an experiment helps understanding its probability. As an example, You’re either going to win in a sports meet, or you’re not.

Other examples:

You get a birthday gift from your sibling, or you don’t.

A medicine cures, or it doesn’t.

You win a lottery ticket, or you don’t.

Key facts to remember:-

The total count of terms in the expansion of (x+y) n are (n+1)

The sum total of exponents of x and y is always n.

nC0, nC1, nC2, nC3, nC4, nC5 … .., nCn are termed as binomial coefficients and is denoted represented by C0, C1, C2, C3, C4, C5, ….., Cn

The binomial coefficients which are at the same distance from the start till the end are equal i.e. nC0 = nCn, nC1 = nCn-1, nC2 = nCn-2 , nC3 = nCn-3 ….etc.

To identify the binomial coefficients we can also use Pascal’s Triangle

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Used in the special cases, if α is a nonnegative integer n, then the (n + 2)th term and all terms ahead in the series are 0, because each consists of a factor (n − n). This section further takes into account the Binomial Theorem and Pascal’s Triangle.

Example1

How to expand (3 + 2x) 6 in ascending powers of x up to the term in x3?

Solution1

The question indicates that we need to apply the general binomial expansions formulas in order to expand the terms in the brackets, but only go as high as x3.

To find the answer, we will substitute 4 for a in the Binomial theorem and 2x for b:

36 + (6C1)(35)(2y) + (6C2)(34)(2x)2 + (6C3)(43)(2x)3 + …

= 729 + (6 ×243 ×2x) + (15 ×81 ×3x2) + (20 ×27 ×8x3) + …

= 729 + 2916x + 3645x2 + 4320x3 + …

The Binomial Theorem for (1 + x)n

This version of the theorem also works when n is any fraction, the binomial theorem becomes: +

(1 + x) n = 1 + nx/1! + n(n-1)2/ 2! + n (n-1) (n-2)x 3/3!+ …..

Note that this form of binomial series goes on forever

Example2

Find the middle term in the expansion of (4a – b) 8.

Solution2

Given that this binomial is raised to the power 8, there are going to be nine terms in the binomial expansion, which makes the 5th term the middle one.

Thus, we will plug 4x, –y, and 8 into the Binomial Theorem,

Considering the number 5 – 1 = 4 as our contrariwise

= 8C4 (4a) 8–4(–b) 4

= (70) (256a4)(b4) = 17920a4b4

Al-Karajī reckoned upon Pascal’s triangle in about 1000 CE

Jia Xian computed Pascal’s triangle up to n = 6 in the mid-11th century.

Isaac Newton discovered the binomial theorem in about 1665 and later asserted, in 1676, without substantial evidence, the general binomial expansion (for any real number n)

The general binomial theorem was stated by John Colson with proof and was published in 1736.

FAQ (Frequently Asked Questions)

Q1. What are the useful Applications of Binomial Theorem?

Ans. Binomial theorem has a huge array of applications in mathematics that includes finding digits of a number, finding the remainder, simplifying a binomial expression etc. Besides, the most common applications of binomial theorem are in algebra. It is commonly used in determining permutations and combinations and probabilities. For exponents of positive integers, n, the theorem was well-recognized to Chinese and Islamic mathematicians of the late medieval period. The binomial theorem can also be generalized to include complex exponents for n, and this was initially evinced by Niels Henrik Abel in the early 19th century.

Q2. What is a Pascal’s Triangle?

Ans. While you already know that

(a+b)^{2} =a^{2} +2ab +b^{2}

and you should be able to figure that

(a+b)^{3} =a^{3} + 3a^{2}b + 3ab^{2} + b^{3}

It must be obvious that (a + b) ¹ = a + b

Thus,

(a + b) ¹ = a + b

(a + b) ² = a² + 2ab + b²

(a + b) ³ = a³ + 3a²b + 3b²a + b³

You should observe that the coefficients of (the numbers before) a and b is:

1 1

1 2 1

1 3 3 1

If you carry on expanding the brackets for higher powers, you will discover that the sequence continues:

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

etc

The aforementioned sequence is called Pascal's triangle. Each of the numbers is identified by directly adding together the two numbers above it.

That said, the number 15 in the last line is found by adding together 10 and 5 above it. Each of the 6s in the last line is found by adding together a 5 and a 1 in the line above. Similarly 10 in the middle line are found by adding 6 and 4 in the first line.

So it is completely feasible to expand (a + b) to any whole number power with the help of Pascal’s triangle.