^{n}

#### RESULTS

The binomial theorem is very helpful in algebra and in addition, to calculate permutations, combinations and probabilities. It can be generalized to add multifaceted exponents for n. Having trouble working out with the Binomial theorem? You’ve come to the right place, our binomial expansion calculator is here to save the day for you. Unlike the theorem itself, our tool is extremely easy to use due to its friendly user interface.

## What is the binomial theorem?

The binomial theorem in the statement is that for any positive number n, the nth power of the totality of two numbers a and b can be articulated as the sum of `\(n + 1\)` relations of the form.

`\(\dbinom{n}{r} a^{n-4}b^e\)`

in the classification of relations (terms), the index r takes on the succeeding values `\(0, 1, 2,..., n\)`. The coefficients, known as the binomial coefficients, are defined by the formula given below:

`\(\dbinom{n}{r} = n!(n-r)!r!\)`

in which `\(n!\)` (n factorial) is the product of the first n natural numbers `\(1, 2, 3,..., n\)` (Note that 0 factorial equals 1). The coefficients can also appear in often referred to as the pascal’s triangle

[img]

by looking for the rth entry in the nth row (counting begins with a zero in either direction). Each entry in the interior of Pascal’s triangle is obtained by adding two entries above it. Hence, the powers of (a + b)n are 1, for `\(n = 0; a + b\)` for `\(n = 1; a2 + 2ab + b2\)` Subsequently fo `\(n = 2; a3 + 3a2b + 3ab2 + b3\)` for `\(n = 3\)` and so on and so forth.

## Binomial theorem formula and Example

`\((a + b)^n = ^nC_0a^n + n^C_1a^n-1b + ^nC_2a^{n-2}b^2 + ^nC_3a^{n-3}b^3 + ... + ^nC_nb^n\)`

The primary example of the binomial theorem is the formula for the square of x+y. The coefficients 1, 2, 1 that appear in this expansion are parallel to the 2nd row of Pascal's triangle. (Because the top "1" of the triangle is row: 0) The coefficients of higher powers of x + y on the other hand correspond to the triangle’s lower rows:

`\((x+y)^3 = x3 + 3x^2y + 3xy^2 + y^3\)`

`\((x+y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\)`

So on and so forth.

## How to use our calculator

The binomial theorem may be tough but using our binomial series calculator just isn’t. Just enter the values required for the purpose of calculation and that’s all you have to do. Leave the math to our tool.

- Enter the value of X and Y
- Enter the value of ‘n’
- Press ‘calculate’

That’s it.

To give you an idea, let’s assume that the value for X and Y are 2 and 3 respectively, while the ‘n’ is 4. If we calculate the binomial theorem using these variables with our calculator, we get:

step #1 (2 + 3) 0 = [1] = 1

step #2 (2 + 3) 1 = [1] 21 30 + [1] 20 31 = 5

step #3 (2 + 3) 2 = [1] 22 30 + [2] 21 31 + [1] 20 32 = 25

step #4 (2 + 3) 3 = [1] 23 30 + [3] 22 31 + [3] 21 32 + [1] 20 33 = 125

step #5 (2 + 3) 4 = [1] 24 30 + [4] 23 31 + [6] 22 32 + [4] 21 33 + [1] 20 34 = 625

625 is our answer: See! That’s how simple it is.

## Some interesting history

The Islamic and Chinese mathematicians of the late medieval era were well-acquainted with this theorem. Al-KarajÄ« determined Pascal’s triangle in 1000 CE, and Jia Xian, in the mid-11th century calculated Pascal’s triangle up to n = 6.