The Binomial Coefficients Calculator will calculate:
- The coefficients of any binomial when both terms and the power of the binomial are given
Binomial Coefficients Calculator Parameters: The power of the binomial is a natural (counting) number.
Binomial Coefficients Calculation and Formula |
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(a + b)^{n} = ^{n}âˆ‘_{k = 0}nk a^{n - k} b^{k} |
Binomial Coefficients Calculator Input Values |
First term of the binomial (a) = |
Second term of the binomial (b) = |
Power of the binomial (n) = |
Please note that the formula for each calculation along with detailed calculations is shown further below this page. As you enter the specific factors of each binomial coefficients calculation, the Binomial Coefficients Calculator will automatically calculate the results and update the formula elements with each element of the binomial coefficients calculation. You can then email or print this binomial coefficients calculation as required for later use.
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The Binomial Coefficients Calculator has practical application and use in the following fields and disciplines
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Theoretical Description
A binomial is a mathematical expression raised to a certain power n where the expression itself contains two terms combined with each other through the operation of addition according to the scheme below.
Binomial = (Term 1 + Term 2)^{n}
The term "binomial" means a "polynomial containing two terms". The power in which a polynomial is raised is called its degree or order. Thus, a binomial raised to the second power is called a second-order (degree) binomial, when raised to the third power it is called a third-order (degree) binomial, etc.
Finding the coefficients of a binomial means finding all numbers preceding the variables in each of the terms when the binomial is written in the decomposed form. In general, we express the terms using the letters x and y (or a and b). For example, in binomials raised to the second power (quadratic binomials), these coefficients are 1, 2 and 1 respectively, because
(a + b)^{2} = a^{2} + 2ab + b^{2}
Likewise, the coefficients in the binomials raised to the third power (cubic binomials) are 1, 3, 3 and 1 respectively, because
(a + b)^{3} = a^{3} + 3a^{2} b + 3ab^{2} + b^{3}
The coefficients of the second and third-order binomials can be found using the expanding brackets method. However, higher order binomials become harder to calculate through the expanding brackets method, so we must use other methods to calculate them. Therefore, we must use a more comprehensive method, which allows the calculation of binomial coefficients of any degree. This method is called the binomial coefficients theorem.
This theorem, first discovered by Sir Isaac Newton, says that the coefficients preceding the variables in binomials raised to a given power are as follows:
(a + b)^{n} = (n0) âˆ™ a^{n} âˆ™ b^{0} + (n1) âˆ™ a^{n - 1} âˆ™ b^{1} + (n2) âˆ™ a^{n - 2} âˆ™ b^{2} + â‹¯ + (nk) âˆ™ a^{n - k} âˆ™ b^{k} + â‹¯ + (nn - 1) âˆ™ a^{1} âˆ™ b^{n - 1} + (nn) âˆ™ a^{0} âˆ™ b^{n}
The general term of this binomial expression therefore is
k^{th} term = (nk) a^{n - k} b^{k}
where
(nk) = n!/k!(n - k)!
The symbol "!" is for "factorial". It means multiplying a number n by all the other numbers from n to 1, i.e. n! = n Ã— (n - 1) Ã— (n - 2) Ã— Ã— [n - (n - 1)]. For example,
5! = 5 Â· 4 Â· 3 Â· 2 Â· 1
8! = 8 Â· 7 Â· 6 Â· 5 Â· 4 Â· 3 Â· 2 Â· 1
etc.
Hence, the algebraic form of expansion of the binomial expression (a + b)^{n} is
(a + b)^{n} = ^{n}âˆ‘_{k = 0}(nk) a^{n - k} b^{k}
where the symbol ^{n}âˆ‘_{k = 0} is an abbreviation that means "the sum of all terms from k = 0 to k = n".
From the above formula and from the definition of factorial, it is clear that the first and the last coefficients are both 1, because
(n0) = C(n,0) = n!/0!(n-0)! = n!/1 âˆ™ n! = 1
and
(nn) = C(n,n) = n!/n!(n - n)! = n!/n! âˆ™ 0! = n!/n! âˆ™ 1 = 1
Look at the two examples below that show how to find the binomial coefficients for n = 6 and for n = 7.
For n = 6, (k varies from 0 to 6) we have
(a + b)^{6}=(60) âˆ™ a^{6} âˆ™ b^{0} + (61) âˆ™ a^{6 - 1} âˆ™ b^{1} + (62) âˆ™ a^{6 - 2} âˆ™ b^{2} + (63) âˆ™ a^{6 - 3} âˆ™ b^{3} + (64) âˆ™ a^{6 - 4} âˆ™ b^{4} + (65) âˆ™ a^{6 - 5} âˆ™ b^{5} + (66) âˆ™ a^{6 - 6} âˆ™ b^{6}
= 6!/0!(6 - 0)! âˆ™ a^{6} âˆ™ b^{0} + 6!/1!(6 - 1)! âˆ™ a^{5} âˆ™ b^{1} + 6!/2!(6 - 2)! âˆ™ a^{4} âˆ™ b^{2} + 6!/3!(6 - 3)! âˆ™ a^{3} âˆ™ b^{3} + 6!/4!(6 - 4)! âˆ™ a^{2} âˆ™ b^{4} + 6!/5!(6 - 5)! âˆ™ a^{1} âˆ™ b^{5} + 6!/6!(6 - 6)! âˆ™ a^{0} âˆ™ b^{6}
= 6!/0! âˆ™ 6! âˆ™ a^{6} âˆ™ b^{0} + 6!/1! âˆ™ 5! âˆ™ a^{5} âˆ™ b^{1} + 6!/2! âˆ™ 4! âˆ™ a^{4} âˆ™ b^{2} + 6!/3! âˆ™ 3! âˆ™ a^{3} âˆ™ b^{3} + 6!/4! âˆ™ 2! âˆ™ a^{2} âˆ™ b^{4} + 6!/5! âˆ™ 1! âˆ™ a^{1} âˆ™ b^{5} + 6!/6! âˆ™ 0! âˆ™ a^{0} âˆ™ b^{6}
= 6!/1 âˆ™ 6! âˆ™ a^{6} âˆ™ 1 + 6 âˆ™ 5!/1 âˆ™ 5! âˆ™ a^{5} âˆ™ b + 6 âˆ™ 5 âˆ™ 4!/2 âˆ™ 1 âˆ™ 4! âˆ™ a^{4} âˆ™ b^{2} + 6 âˆ™ 5 âˆ™ 4 âˆ™ 3!/3 âˆ™ 2 âˆ™ 1 âˆ™ 3! âˆ™ a^{3} âˆ™ b^{3} + 6 âˆ™ 5 âˆ™ 4!/4! âˆ™ 2 âˆ™ 1 âˆ™ a^{2} âˆ™ b^{4} + 6 âˆ™ 5!/5! âˆ™ 1 âˆ™ a^{1} âˆ™ b^{5} + 6!/6! âˆ™ 1 âˆ™ 1 âˆ™ b^{6}
= a^{6} + 6a^{5} b + 15a^{4} b^{2} + 20a^{3} b^{3} + 15a^{2} b^{4} + 6ab^{5} + b^{6}
For n = 7 (k varies from 0 to 7), we have
(a + b)^{7}=(70) âˆ™ a^{7} âˆ™ b^{0} + (71) âˆ™ a^{7 - 1} âˆ™ b^{1} + (72) âˆ™ a^{7 - 2} âˆ™ b^{2} + (73) âˆ™ a^{7 - 3} âˆ™ b^{3} + (74) âˆ™ a^{7 - 4} âˆ™ b^{4} + (75) âˆ™ a^{7 - 5} âˆ™ b^{5} + (76) âˆ™ a^{7 - 6} âˆ™ b^{6} + (77) âˆ™ a^{7 - 7} âˆ™ b^{7}
= 7!/0!(7 - 0)! âˆ™ a^{7} âˆ™ b^{0} + 7!/1!(7 - 1)! âˆ™ a^{6} âˆ™ b^{1} + 7!/2!(7 - 2)! âˆ™ a^{5} âˆ™ b^{2} + 7!/3!(7 - 3)! âˆ™ a^{4} âˆ™ b^{3} + 7!/4!(7 - 4)! âˆ™ a^{3} âˆ™ b^{4} + 7!/5!(7 - 5)! âˆ™ a^{2} âˆ™ b^{5} + 7!/6!(7 - 6)! âˆ™ a^{1} âˆ™ b^{6} + 7!/7!(7 - 7)! âˆ™ a^{0} âˆ™ b^{7}
= 7!/0! âˆ™ 7! âˆ™ a^{7} âˆ™ b^{0} + 7!/1! âˆ™ 6! âˆ™ a^{6} âˆ™ b^{1} + 7!/2! âˆ™ 5! âˆ™ a^{5} âˆ™ b^{2} + 7!/3! âˆ™ 4! âˆ™ a^{4} âˆ™ b^{3} + 7!/4! âˆ™ 3! âˆ™ a^{3} âˆ™ b^{4} + 7!/5! âˆ™ 2! âˆ™ a^{2} âˆ™ b^{5} + 7!/6! âˆ™ 1! âˆ™ a^{1} âˆ™ b^{6} + 7!/7! âˆ™ 0! âˆ™ a^{0} âˆ™ b^{7}
= 7!/1 âˆ™ 7! âˆ™ a^{7} âˆ™ 1 + 7 âˆ™ 6!/1! âˆ™ 6! âˆ™ a^{6} âˆ™ b + 7 âˆ™ 6 âˆ™ 5!/2 âˆ™ 1 âˆ™ 5! âˆ™ a^{5} âˆ™ b^{2} + 7 âˆ™ 6 âˆ™ 5 âˆ™ 4!/3 âˆ™ 2 âˆ™ 1 âˆ™ 4! âˆ™ a^{4} âˆ™ b^{3} + 7 âˆ™ 6 âˆ™ 5 âˆ™ 4!/4! âˆ™ 3 âˆ™ 2 âˆ™ 1 âˆ™ a^{3} âˆ™ b^{4} + 7 âˆ™ 6 âˆ™ 5!/5! âˆ™ 2 âˆ™ 1 âˆ™ a^{2} âˆ™ b^{5} + 7 âˆ™ 6!/6! âˆ™ 1 âˆ™ a âˆ™ b^{6} + 7!/7! âˆ™ 1 âˆ™ 1 âˆ™ b^{7}
= a^{7} + 7a^{6} b + 21a^{5} b^{2} + 35a^{4} b^{3} + 35a^{3} b^{4} + 21a^{2} b^{5} + 7ab^{6} + b^{7}
Instructions and information for using the Binomial Coefficients Calculator
All you have to do is to insert the two terms a and b of the binomial as well as the index (exponent) n that shows the power of the binomial. The terms a and b are not meant to be just single letters; they can also be monomials. For example, you may insert 2x for a and 3y for b. The calculator will eventually list all the binomial coefficients when the original binomial is written in the disassembled form.
For example, if you insert 2x for a, 4y for b and 3 for n, the calculator gives the following coefficients:
Output 1 = 8; Output 2 = 48; Output 3 = 96; Output 4 = 64
because
(a + b)^{3} = (30) âˆ™ a^{3} âˆ™ b^{0} + (31) âˆ™ a^{3 - 1} âˆ™ b^{1} + (32) âˆ™ a^{3 - 2} âˆ™ b^{2} + (33) âˆ™ a^{3 - 3} âˆ™ b^{3}
= 3!/0!(3-0)! âˆ™ a^{3} âˆ™ b^{0} + 3!/1!(3-1)! âˆ™ a^{2} âˆ™ b^{1} + 3!/2!(3-2)! âˆ™ a^{1} âˆ™ b^{2} + 3!/3!(3 - 3)! âˆ™ a^{0} âˆ™ b^{3}
= 3!/1! âˆ™ 3! âˆ™ a^{3} + 3!/1! âˆ™ 2! âˆ™ a^{2} b + 3!/2! âˆ™ 1! âˆ™ ab^{2} + 3!/3! âˆ™ 0! âˆ™ b^{3}
= a^{3} + 3a^{2} b + 3ab^{2} + b^{3}
= (2x)^{3} + 3 âˆ™ (2x)^{2} âˆ™ (4y) + 3 âˆ™ (2x) âˆ™ (4y)^{2} + (4y)^{3}
= 8x^{3} + 3 âˆ™ 4x^{2} âˆ™ 4y + 3 âˆ™ 2x âˆ™ 16y^{2} + 64y^{3}
= 8x^{3} + 48x^{2} y + 96xy^{2} + 64y^{3}
Remark! The terms a and b can be negative as well. All the above operations are carried out in the same way as when both terms are positive but by taking into account the changes determined by any term's sign.
Sequences and Series Math Tutorials associated with the Binomial Coefficients Calculator
The following Math tutorials are provided within the Sequences and Series section of our Free Math Tutorials. Each Sequences and Series tutorial includes detailed Sequences and Series formula and example of how to calculate and resolve specific Sequences and Series questions and problems. At the end of each Sequences and Series tutorial you will find Sequences and Series revision questions with a hidden answer that reveal when clicked. This allows you to learn about Sequences and Series and test your knowledge of Math by answering the revision questions on Sequences and Series.
- 12.1 - Working with Term-to-Term Rules in Sequences
- 12.2 - Working with Arithmetic and Geometric Series. How to find the Sum of the First n-Terms of a Series.
- 12.3 - Binomial Expansion and Coefficients
- 12.4 - Infinite Series Explained
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