Binomial theorem explained

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form, where the exponents and are nonnegative integers satisfying and the coefficient of each term is a specific positive integer depending on and . For example, for,$$(x+y)^4\; =\; x^4\; +\; 4\; x^3y\; +\; 6\; x^2\; y^2\; +\; 4\; x\; y^3\; +\; y^4.$$

The coefficient in each term is known as the binomial coefficient or (the two have the same value). These coefficients for varying and can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where gives the number of different combinations (i.e. subsets) of elements that can be chosen from an -element set. Therefore is usually pronounced as " choose ".

Statement

According to the theorem, the expansion of any nonnegative integer power of the binomial is a sum of the form$$(x+y)^n\; =\; x^n\; y^0\; +\; x^\; y^1\; +\; x^\; y^2\; +\; \backslash cdots\; +\; x^0\; y^n,$$where each

is a positive integer known as a binomial coefficient, defined as

$$\backslash binom\; nk\; =\; \backslash frac\; =\; \backslash frac.$$

This formula is also referred to as the binomial formula or the binomial identity. Using summation notation, it can be written more concisely as$$(x+y)^n\; =\; \backslash sum\_^n\; x^y^k\; =\; \backslash sum\_^n\; x^y^.$$

The final expression follows from the previous one by the symmetry of and in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetric, $\backslash binom\; nk\; =\; \backslash binom\; n.$^[1]

A simple variant of the binomial formula is obtained by substituting for, so that it involves only a single variable. In this form, the formula reads

Examples

The first few cases of the binomial theorem are:In general, for the expansion of on the right side in the th row (numbered so that the top row is the 0th row):

• the exponents of in the terms are (the last term implicitly contains);
• the exponents of in the terms are (the first term implicitly contains);
• the coefficients form the th row of Pascal's triangle;
• before combining like terms, there are terms in the expansion (not shown);
• after combining like terms, there are terms, and their coefficients sum to .

An example illustrating the last two points: with

.

A simple example with a specific positive value of :

A simple example with a specific negative value of :

Geometric explanation

For positive values of and, the binomial theorem with is the geometrically evident fact that a square of side can be cut into a square of side, a square of side, and two rectangles with sides and . With, the theorem states that a cube of side can be cut into a cube of side, a cube of side, three rectangular boxes, and three rectangular boxes.

^[2] if one sets and interpreting as an infinitesimal change in, then this picture shows the infinitesimal change in the volume of an -dimensional hypercube, where the coefficient of the linear term (in ) is the area of the faces, each of dimension :$$(x+\backslash Delta\; x)^n\; =\; x^n\; +\; nx^\backslash Delta\; x\; +\; \backslash binomx^(\backslash Delta\; x)^2\; +\; \backslash cdots.$$Substituting this into the definition of the derivative via a difference quotient and taking limits means that the higher order terms, and higher, become negligible, and yields the formula interpreted as "the infinitesimal rate of change in volume of an -cube as side length varies is the area of of its -dimensional faces". If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.

Binomial coefficients

See main article: Binomial coefficient. The coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written

and pronounced " choose ".

Formulas

The coefficient of is given by the formula$$\backslash binom\; =\; \backslash frac,$$which is defined in terms of the factorial function . Equivalently, this formula can be written$$\backslash binom\; =\; \backslash frac\; =\; \backslash prod\_^k\; \backslash frac\; =\; \backslash prod\_^\; \backslash frac$$with factors in both the numerator and denominator of the fraction. Although this formula involves a fraction, the binomial coefficient

is actually an integer.

Combinatorial interpretation

The binomial coefficient

can be interpreted as the number of ways to choose elements from an -element set (a combination). This is related to binomials for the following reason: if we write as a product$$(x+y)(x+y)(x+y)\backslash cdots(x+y),$$then, according to the distributive law, there will be one term in the expansion for each choice of either or from each of the binomials of the product. For example, there will only be one term, corresponding to choosing from each binomial. However, there will be several terms of the form, one for each way of choosing exactly two binomials to contribute a . Therefore, after combining like terms, the coefficient of will be equal to the number of ways to choose exactly elements from an -element set.

Proofs

Combinatorial proof

Expanding yields the sum of the products of the form where each is or . Rearranging factors shows that each product equals for some between and . For a given, the following are proved equal in succession:

• the number of terms equal to in the expansion
• the number of -character strings having in exactly positions
• the number of -element subsets of

either by definition, or by a short combinatorial argument if one is defining as This proves the binomial theorem.

Example

The coefficient of inequals

because there are three strings of length 3 with exactly two 's, namely,$$xyy,\; \backslash ;\; yxy,\; \backslash ;\; yyx,$$corresponding to the three 2-element subsets of, namely,$$\backslash ,\backslash ;\backslash ,\backslash ;\backslash ,$$where each subset specifies the positions of the in a corresponding string.

Inductive proof

Induction yields another proof of the binomial theorem. When, both sides equal, since and

Now suppose that the equality holds for a given ; we will prove it for . For, let denote the coefficient of in the polynomial . By the inductive hypothesis, is a polynomial in and such that is if, and otherwise. The identity$$(x+y)^\; =\; x(x+y)^n\; +\; y(x+y)^n$$shows that is also a polynomial in and, and$$[(x+y)^\{n+1\}]\_\; =\; [(x+y)^n]\_\; +\; [(x+y)^n]\_,$$since if, then and . Now, the right hand side is$$\backslash binom\; +\; \backslash binom\; =\; \backslash binom,$$by Pascal's identity.^[3] On the other hand, if, then and, so we get . Thus$$(x+y)^\; =\; \backslash sum\_^\; \backslash binom\; x^\; y^k,$$which is the inductive hypothesis with substituted for and so completes the inductive step.

Generalizations

Generalized binomial theorem

See main article: Binomial series.

The standard binomial theorem, as discussed above, is concerned with

where the exponent is a nonnegative integer. The generalized binomial theorem allows for non-integer, negative, or even complex exponents, at the expense of replacing the finite sum by an infinite series.

In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper index, which cannot be done using the usual formula with factorials. However, for an arbitrary number, one can define$$=\backslash frac\; =\backslash frac,$$where the last equation introduces modern notation for the falling factorial. This agrees with the usual definitions when is a nonnegative integer. Then, if and are real numbers with,^[4] and is any complex number, one has

When is a nonnegative integer, the binomial coefficients for are zero, so this equation reduces to the usual binomial theorem, and there are at most nonzero terms. For other values of, the series has infinitely many nonzero terms.

For example, gives the following series for the square root:$$\backslash sqrt\; =\; 1\; +\; \backslash fracx\; -\; \backslash fracx^2\; +\; \backslash fracx^3\; -\; \backslash fracx^4\; +\; \backslash fracx^5\; -\; \backslash cdots.$$

With, the generalized binomial series becomes:$$(1+x)^\; =\; \backslash frac\; =\; 1\; -\; x\; +\; x^2\; -\; x^3\; +\; x^4\; -\; x^5\; +\; \backslash cdots.$$ which is the geometric series sum formula for the convergent case, whose common ratio is .

More generally, with, we have for :^[5] $$\backslash frac\; =\; \backslash sum\_^\backslash infty\; x^k\; =\; \backslash sum\_^\backslash infty\; (-1)^k\; x^k.$$

So, for instance, when,$$\backslash frac\; =\; 1\; -\; \backslash fracx\; +\; \backslash fracx^2\; -\; \backslash fracx^3\; +\; \backslash fracx^4\; -\; \backslash fracx^5\; +\; \backslash cdots.$$

Replacing with yields:$$\backslash frac\; =\; \backslash sum\_^\backslash infty\; (-1)^k\; (-x)^k\; =\; \backslash sum\_^\backslash infty\; x^k.$$

So, for instance, when, we have for :$$\backslash frac\; =\; 1\; +\; \backslash fracx\; +\; \backslash fracx^2\; +\; \backslash fracx^3\; +\; \backslash fracx^4\; +\; \backslash fracx^5\; +\; \backslash cdots.$$

Further generalizations

The generalized binomial theorem can be extended to the case where and are complex numbers. For this version, one should again assume and define the powers of and using a holomorphic branch of log defined on an open disk of radius centered at . The generalized binomial theorem is valid also for elements and of a Banach algebra as long as, and is invertible, and .

A version of the binomial theorem is valid for the following Pochhammer symbol-like family of polynomials: for a given real constant, define

and$$x^\; =\; \backslash prod\_^[x+(k-1)c]$$for Then^[6] $$(a\; +\; b)^\; =\; \backslash sum\_^\backslash binoma^b^.$$The case recovers the usual binomial theorem.

More generally, a sequence

of polynomials is said to be of binomial type if for all , , and for all , , and .An operator on the space of polynomials is said to be the basis operator of the sequence if and for all . A sequence is binomial if and only if its basis operator is a Delta operator.^[7] Writing for the shift by operator, the Delta operators corresponding to the above "Pochhammer" families of polynomials are the backward difference for , the ordinary derivative for , and the forward difference for .

Multinomial theorem

See main article: Multinomial theorem.

The binomial theorem can be generalized to include powers of sums with more than two terms. The general version is

$$(x\_1\; +\; x\_2\; +\; \backslash cdots\; +\; x\_m)^n\; =\; \backslash sum\_\; \backslash binom\; x\_1^\; x\_2^\; \backslash cdots\; x\_m^,$$

where the summation is taken over all sequences of nonnegative integer indices through such that the sum of all is . (For each term in the expansion, the exponents must add up to ). The coefficients

are known as multinomial coefficients, and can be computed by the formula$$\backslash binom\; =\; \backslash frac.$$

Combinatorially, the multinomial coefficient

counts the number of different ways to partition an -element set into disjoint subsets of sizes .

Multi-binomial theorem

When working in more dimensions, it is often useful to deal with products of binomial expressions. By the binomial theorem this is equal to$$(x\_1+y\_1)^\backslash dotsm(x\_d+y\_d)^\; =\; \backslash sum\_^\backslash dotsm\backslash sum\_^\; \backslash binom\; x\_1^y\_1^\; \backslash dotsc\; \backslash binom\; x\_d^y\_d^.$$

This may be written more concisely, by multi-index notation, as$$(x+y)^\backslash alpha\; =\; \backslash sum\_\; \backslash binom\; x^\backslash nu\; y^.$$

General Leibniz rule

See main article: General Leibniz rule.

The general Leibniz rule gives the th derivative of a product of two functions in a form similar to that of the binomial theorem:^[8] $$(fg)^(x)\; =\; \backslash sum\_^n\; \backslash binom\; f^(x)\; g^(x).$$

Here, the superscript indicates the th derivative of a function,

. If one sets and, cancelling the common factor of from each term gives the ordinary binomial theorem.^[9]

History

Special cases of the binomial theorem were known since at least the 4th century BC when Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent

.^[10] Greek mathematician Diophantus cubed various binomials, including . Indian mathematician Aryabhata's method for finding cube roots, from around 510 AD, suggests that he knew the binomial formula for exponent .

Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting objects out of without replacement (combinations), were of interest to ancient Indian mathematicians. The Jain Bhagavati Sutra (c. 300 BC) describes the number of combinations of philosophical categories, senses, or other things, with correct results up through (probably obtained by listing all possibilities and counting them)^[11] and a suggestion that higher combinations could likewise be found.^[12] The Chandaḥśāstra by the Indian lyricist Piṅgala (3rd or 2nd century BC) somewhat cryptically describes a method of arranging two types of syllables to form metres of various lengths and counting them; as interpreted and elaborated by Piṅgala's 10th-century commentator Halāyudha his "method of pyramidal expansion" (meru-prastāra) for counting metres is equivalent to Pascal's triangle.^[13] Varāhamihira (6th century AD) describes another method for computing combination counts by adding numbers in columns.^[14] By the 9th century at latest Indian mathematicians learned to express this as a product of fractions, and clear statements of this rule can be found in Śrīdhara's Pāṭīgaṇita (8th–9th century), Mahāvīra's Gaṇita-sāra-saṅgraha (c. 850), and Bhāskara II's Līlāvatī (12th century).^[15]

The Persian mathematician al-Karajī (953–1029) wrote a now-lost book containing the binomial theorem and a table of binomial coefficients, often credited as their first appearance.^{[16] [17] [18] [19]} An explicit statement of the binomial theorem appears in al-Samawʾal's al-Bāhir (12th century), there credited to al-Karajī. Al-Samawʾal algebraically expanded the square, cube, and fourth power of a binomial, each in terms of the previous power, and noted that similar proofs could be provided for higher powers, an early form of mathematical induction. He then provided al-Karajī's table of binomial coefficients (Pascal's triangle turned on its side) up to and a rule for generating them equivalent to the recurrence relation . The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, although many of his mathematical works are lost. The binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui^[20] and also Chu Shih-Chieh. Yang Hui attributes the method to a much earlier 11th century text of Jia Xian, although those writings are now also lost.^[21]

In Europe, descriptions of the construction of Pascal's triangle can be found as early as Jordanus de Nemore's De arithmetica (13th century).^[22] In 1544, Michael Stifel introduced the term "binomial coefficient" and showed how to use them to express

in terms of , via "Pascal's triangle".^[23] Other 16th century mathematicians including Niccolò Fontana Tartaglia and Simon Stevin also knew of it. 17th-century mathematician Blaise Pascal studied the eponymous triangle comprehensively in his Traité du triangle arithmétique.^[24]

The development of the binomial theorem for positive integer exponents is attributed to Al-Kashi by the year 1427. The first proper proof of the binomial theorem for positive integral index was given by Pascal.^[25] By the early 17th century, some specific cases of the generalized binomial theorem, such as for

, can be found in the work of Henry Briggs' Arithmetica Logarithmica (1624). Isaac Newton discovered the generalized binomial theorem, valid for any real exponent, in 1664-5, inspired by the work of John Wallis's Arithmetic Infinitorum and his method of interpolation.^{[26] [27] [28]} A logarithmic version of the theorem for fractional exponents was discovered independently by James Gregory who wrote down his formula in 1670.^[29]

Applications

Multiple-angle identities

For the complex numbers the binomial theorem can be combined with de Moivre's formula to yield multiple-angle formulas for the sine and cosine. According to De Moivre's formula,$$\backslash cos\backslash left(nx\backslash right)+i\backslash sin\backslash left(nx\backslash right)\; =\; \backslash left(\backslash cos\; x+i\backslash sin\; x\backslash right)^n.$$

Using the binomial theorem, the expression on the right can be expanded, and then the real and imaginary parts can be taken to yield formulas for and . For example, since$$\backslash left(\backslash cos\; x\; +\; i\backslash sin\; x\backslash right)^2\; =\; \backslash cos^2\; x\; +\; 2i\; \backslash cos\; x\; \backslash sin\; x\; -\; \backslash sin^2\; x=\; (\backslash cos^2\; x-\backslash sin^2\; x)\; +\; i(2\backslash cos\; x\backslash sin\; x),$$But De Moivre's formula identifies the left side with

, so$$\backslash cos(2x)\; =\; \backslash cos^2\; x\; -\; \backslash sin^2\; x\; \backslash quad\backslash text\backslash quad\backslash sin(2x)\; =\; 2\; \backslash cos\; x\; \backslash sin\; x,$$which are the usual double-angle identities. Similarly, since$$\backslash left(\backslash cos\; x\; +\; i\backslash sin\; x\backslash right)^3\; =\; \backslash cos^3\; x\; +\; 3i\; \backslash cos^2\; x\; \backslash sin\; x\; -\; 3\; \backslash cos\; x\; \backslash sin^2\; x\; -\; i\; \backslash sin^3\; x,$$De Moivre's formula yields$$\backslash cos(3x)\; =\; \backslash cos^3\; x\; -\; 3\; \backslash cos\; x\; \backslash sin^2\; x\; \backslash quad\backslash text\backslash quad\; \backslash sin(3x)\; =\; 3\backslash cos^2\; x\; \backslash sin\; x\; -\; \backslash sin^3\; x.$$In general,$$\backslash cos(nx)\; =\; \backslash sum\_\; (-1)^\; \backslash cos^\; x\; \backslash sin^k\; x$$and$$\backslash sin(nx)\; =\; \backslash sum\_\; (-1)^\; \backslash cos^\; x\; \backslash sin^k\; x.$$There are also similar formulas using Chebyshev polynomials.

Series for e

The number is often defined by the formula$$e\; =\; \backslash lim\_\; \backslash left(1\; +\; \backslash frac\backslash right)^n.$$

Applying the binomial theorem to this expression yields the usual infinite series for . In particular:$$\backslash left(1\; +\; \backslash frac\backslash right)^n\; =\; 1\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots\; +\; \backslash frac.$$

The th term of this sum is$$\backslash frac\; =\; \backslash frac\backslash cdot\backslash frac$$

As, the rational expression on the right approaches, and therefore$$\backslash lim\_\; \backslash frac\; =\; \backslash frac.$$

This indicates that can be written as a series:$$e=\backslash sum\_^\backslash infty\backslash frac=\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots.$$

Indeed, since each term of the binomial expansion is an increasing function of, it follows from the monotone convergence theorem for series that the sum of this infinite series is equal to .

Probability

The binomial theorem is closely related to the probability mass function of the negative binomial distribution. The probability of a (countable) collection of independent Bernoulli trials

with probability of success all not happening is$$P\backslash biggl(\backslash bigcap\_\; X\_t^C\backslash biggr)\; =\; (1-p)^$$ = \sum_^ (-p)^n.An upper bound for this quantity is ^[30]

In abstract algebra

The binomial theorem is valid more generally for two elements and in a ring, or even a semiring, provided that . For example, it holds for two matrices, provided that those matrices commute; this is useful in computing powers of a matrix.^[31]

The binomial theorem can be stated by saying that the polynomial sequence is of binomial type.

See also

• Binomial approximation
• Binomial distribution
• Binomial inverse theorem
• Binomial coefficient
• Stirling's approximation
• Tannery's theorem
• Polynomials calculating sums of powers of arithmetic progressions
• q-binomial theorem

Further reading

• Book: Graham . Ronald . Ronald Graham . Knuth . Donald . Donald Knuth . Oren . Patashnik . Oren Patashnik . . Addison Wesley . 1994 . 2nd . Ch. 5, . Binomial Coefficients . 978-0-201-55802-9 .

External links

• • Binomial Theorem by Stephen Wolfram, and "Binomial Theorem (Step-by-Step)" by Bruce Colletti and Jeff Bryant, Wolfram Demonstrations Project, 2007.

Notes and References

1. $(x+y)^n\; =\; \backslash sum\_^n\; x^y^k\; =\; \backslash sum\_^n\; x^y^$, and the coefficient of the same monomial in the left and right-hand side expressions of the 2nd equality must be same; for $x^y^k\; =\; x^y^$ so $k\text{'}\; =\; n\; -\; k$, $\backslash binom\; =\; \backslash binom\; =\; \backslash binom$.
2. Barth . Nils R.. Computing Cavalieri's Quadrature Formula by a Symmetry of the n-Cube . 10.2307/4145193 . 4145193 . The American Mathematical Monthly . 111. 9. 811–813 . 2004.
3. http://proofs.wiki/Binomial_theorem Binomial theorem
4. This is to guarantee convergence. Depending on, the series may also converge sometimes when .
5. Web site: Negative Binomial Series. Wolfram MathWorld. Weisstein. Eric W..
6. Problem 352. Dan. Sokolowsky. Basil C.. Rennie. Crux Mathematicorum. 5. 2. 1979 . 55–56.
7. Book: Aigner, Martin . Martin Aigner . Combinatorial Theory . limited . 1979 . Springer . 0-387-90376-3 . 105 .
8. Book: Olver, Peter J. . Peter J. Olver . 2000 . Applications of Lie Groups to Differential Equations . Springer . 318–319 . 9780387950006 .
9. Book: Spivey . Michael Z. . The Art of Proving Binomial Identities . 2019 . CRC Press . 978-1351215800 . 71.
10. The Story of the Binomial Theorem. J. L.. Coolidge. The American Mathematical Monthly. 56. 3. 1949. 147–157. 10.2307/2305028. 2305028.
11. Biggs . Norman L. . Norman L. Biggs . The roots of combinatorics . Historia Mathematica . 6 . 1979 . 2 . 109–136 . 10.1016/0315-0860(79)90074-0 . free.
12. Datta . Bibhutibhushan . Bibhutibhushan Datta . The Jaina School of Mathematics . Bulletin of the Calcutta Mathematical Society . 27 . 1929 . 5. 115–145 (esp. 133–134) . Reprinted as "The Mathematical Achievements of the Jainas" in Book: Chattopadhyaya . Debiprasad . Studies in the History of Science in India . 2 . New Delhi . Editorial Enterprises . 1982 . 684–716.
13. Bag . Amulya Kumar . Binomial theorem in ancient India . Indian Journal of History of Science . 1 . 1 . 1966 . 68–74 . Shah . Jayant . 2013 . Gaṇita Bhāratī . 35 . 1–4 . 43–96 . A History of Piṅgala's Combinatorics . . (Preprint) Survey sources: Book: Edwards, A. W. F. . A. W. F. Edwards . 1987 . The combinatorial numbers in India . Pascal's Arithmetical Triangle . London . Charles Griffin . 0-19-520546-4 . https://archive.org/details/pascalsarithmeti0000edwa/page/27 . 27–33 . limited . Book: Divakaran, P. P. . 2018 . The Mathematics of India: Concepts, Methods, Connections . Combinatorics . §5.5 . Springer; Hindustan Book Agency . 10.1007/978-981-13-1774-3_5 . 978-981-13-1773-6 . Book: Roy, Ranjan . Ranjan Roy . 2021 . Series and Products in the Development of Mathematics . 2 . 1 . Cambridge University Press . The Binomial Theorem . Ch. 4, . 978-1-108-70945-3 . 10.1017/9781108709453.005 .
14. Gupta . Radha Charan . Radha Charan Gupta . Varāhamihira's Calculation of and the Discovery of Pascal's Triangle . Gaṇita Bhāratī . 14 . 1–4 . 1992 . 45–49 . Reprinted in Book: Ramasubramanian . K. . 2019 . Gaṇitānanda . Springer . 10.1007/978-981-13-1229-8_29 . 285–289 .
15. Book: 1959. The Patiganita of Sridharacarya . Shukla . Kripa Shankar . Kripa Shankar Shukla . Lucknow University . https://archive.org/details/Patiganita/page/n294/mode/1up . Combinations of Savours . Vyavahāras 1.9, (text), (translation) .
16. Yadegari . Mohammad . 1980 . The Binomial Theorem: A Widespread Concept in Medieval Islamic Mathematics . Historia Mathematica . 7 . 4 . 401–406 . 10.1016/0315-0860(80)90004-X . free .
17. Rashed . Roshdi . Roshdi Rashed . 1972 . L'induction mathématique: al-Karajī, al-Samawʾal . Archive for History of Exact Sciences . 9 . 1 . 1–21 . 41133347 . 10.1007/BF00348537 . fr . Translated into English by A. F. W. Armstrong in Book: Rashed, Roshdi . 1994 . The Development of Arabic Mathematics: Between Arithmetic and Algebra . Mathematical Induction: al-Karajī and al-Samawʾal . https://archive.org/details/RoshdiRashedauth.TheDevelopmentOfArabicMathematicsBetweenArithmeticAndAlgebraSpringerNetherlands1994/page/n71/ . §1.4, . 10.1007/978-94-017-3274-1_2 . Kluwer . 0-7923-2565-6 . "The first formulation of the binomial and the table of binomial coefficients, to our knowledge, is to be found in a text by al-Karajī, cited by al-Samawʾal in al-Bāhir." .
18. Encyclopedia: Al-Karajī . Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures . Sesiano . Jacques . Selin . Helaine . Helaine Selin . 1997 . Springer . 10.1007/978-94-017-1416-7_11 . 978-94-017-1418-1 . 475–476 . Another [lost work of Karajī's] contained the first known explanation of the arithmetical (Pascal's) triangle; the passage in question survived through al-Samawʾal's Bāhir (twelfth century) which heavily drew from the Badīʿ. .
19. Berggren . John Lennart . 1985 . History of mathematics in the Islamic world: The present state of the art . Review of Middle East Studies . 19 . 1 . 9–33 . 10.1017/S0026318400014796 . Republished in Book: From Alexandria, Through Baghdad . Sidoli . Nathan . Brummelen . Glen Van . Glen Van Brummelen . 2014 . Springer . 978-3-642-36735-9 . 10.1007/978-3-642-36736-6_4 . 51–71 . [...] since the table of binomial coefficients had been previously found in such late works as those of al-Kāshī (fifteenth century) and Naṣīr al-Dīn al-Ṭūsī (thirteenth century), some had suggested that the table was a Chinese import. However, the use of the binomial coefficients by Islamic mathematicians of the eleventh century, in a context which had deep roots in Islamic mathematics, suggests strongly that the table was a local discovery – most probably of al-Karajī..
20. Web site: Landau . James A. . Historia Matematica Mailing List Archive: Re: [HM] Pascal's Triangle ]. Archives of Historia Matematica . mailing list email . 2007-04-13 . 1999-05-08 . 2021-02-24 . https://web.archive.org/web/20210224081637/http://archives.math.utk.edu/hypermail/historia/may99/0073.html . dead .
21. Book: Martzloff, Jean-Claude . A History of Chinese Mathematics . Jia Xian and Liu Yi . Wilson . Stephen S. . Springer . 1997 . French ed. 1987 . 3-540-54749-5 . 142 . https://archive.org/details/historyofchinese0000mart_g2q8/page/142/mode/2up?&q=%22depends+on+the+binomial+expansion%22 . limited .
22. Hughes . Barnabas. 1989 . The arithmetical triangle of Jordanus de Nemore . Historia Mathematica . 16 . 3 . 213–223 . 10.1016/0315-0860(89)90018-9 . free .
23. Book: Kline, Morris . Morris Kline . Mathematical Thought From Ancient to Modern Times . Oxford University Press . 1972 . 273 . . 77-170263 .
24. Book: Katz, Victor . Victor Katz . A History of Mathematics: An Introduction . 3rd . Addison-Wesley . 2009 . 1993 . 978-0-321-38700-4 . § 14.3, . Elementary Probability .
25. Book: Edwards, A. W. F. . A. W. F. Edwards . Pascal's Arithmetical Triangle: The Story of a Mathematical Idea . 2002-07-23 . JHU Press . 978-0-8018-6946-4 . 55, 79 . en.
26. Book: Edwards, C. Henry . The Historical Development of the Calculus . 1994 . Springer-Verlag . 978-0-387-94313-8 . 167–169, 178–187 . 10.1007/978-1-4612-6230-5.
27. Whiteside . D. T. . Tom Whiteside . 1961 . Newton's Discovery of the General Binomial Theorem . The Mathematical Gazette . en . 45 . 353 . 175–180 . 10.2307/3612767 . 3612767 . subscription .
28. Book: Iliffe . Rob . The Cambridge Companion to Newton . Smith . George Edwin . 2016 . Cambridge University Press . 978-1-107-60174-1 . 2nd . Cambridge . 389–390.
29. Book: Stillwell, John . John Stillwell . Mathematics and its history . 2010 . Springer . 978-1-4419-6052-8 . 186 . 3rd.
30. Book: Elements of Information Theory . Data Compression . Cover . Thomas M. . Thomas M. Cover . Thomas . Joy A. . Joy A. Thomas . 1991 . Wiley . 9780471062592 . Ch. 5, . 10.1002/0471200611.ch5.
31. Book: Artin, Michael . Michael Artin . Algebra . 2nd . 2011 . Pearson . equation (4.7.11).