# Binomial theorem
The binomial theorem is the formula for expanding a power of a sum: for any natural number \( n \) and real or complex numbers \( x \) and \( y \),
\[ (x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^{k} y^{n-k}, \]
where \( \binom{n}{k} \), read "\( n \) choose \( k \)", is the binomial coefficient \( n!/(k!(n-k)!) \) for \( 0 \leqslant k \leqslant n \), with the convention \( \binom{n}{k} = 0 \) when \( k > n \)[1](https://courses.maths.ox.ac.uk/mod/page/view.php?id=66328). The theorem gives the coefficient of each term in the expansion and, through its coefficients, connects algebra with counting, probability, and infinite series.
| Key fact | Detail |
|---|---|
| Statement | \( (x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^{k} y^{n-k} \) for natural \( n \) and real or complex \( x, y \)[1](https://courses.maths.ox.ac.uk/mod/page/view.php?id=66328) |
| Coefficients | \( \binom{n}{k} = n!/(k!(n-k)!) \), the number of ways of choosing \( k \) elements from a set of size \( n \)[1](https://courses.maths.ox.ac.uk/mod/page/view.php?id=66328) |
| Pascal's rule | \( \binom{n}{k-1} + \binom{n}{k} = \binom{n+1}{k} \), the recurrence that builds Pascal's triangle[1](https://courses.maths.ox.ac.uk/mod/page/view.php?id=66328) |
| Earliest known work | Attributed to al-Karaji (ca. 1029); an expansion up to the seventh power appears in a work of al-Zanjani (d. 1262)[2](https://www.sciencedirect.com/science/article/pii/031508608090004X) |
| Newton's extension | For real \( r \) not a nonnegative integer, \( (1+x)^{r} = \sum_{i=0}^{\infty} \binom{r}{i} x^{i} \) converges for \( -1 < x < 1 \)[3](https://www.whitman.edu/mathematics/cgt_online/book/section03.01.html) |
| Probability use | The binomial distribution \( p_N(N_A) = \binom{N}{N_A} p_A^{N_A} p_B^{N-N_A} \) takes its prefactor directly from the expansion of \( (p_A + p_B)^N \)[4](https://web.mit.edu/8.Math/www/lectures/lec15/4.1.3.pdf) |
## Binomial coefficients and Pascal's triangle
The coefficient \( \binom{n}{k} \) counts the ways of choosing \( k \) elements from a set of size \( n \), and is also written \( ^{n}C_{k} \)[1](https://courses.maths.ox.ac.uk/mod/page/view.php?id=66328). Although the factorial formula produces rational numbers on its face, a short induction using Pascal's rule shows that every binomial coefficient is an integer[1](https://courses.maths.ox.ac.uk/mod/page/view.php?id=66328).
Laid out in rows, the coefficients form [Pascal's triangle](https://www.edgepedia.org/pascals-triangle), in which each entry is the sum of the two entries most immediately above it; the \( n \)th row contains the non-zero \( \binom{n}{k} \)[1](https://courses.maths.ox.ac.uk/mod/page/view.php?id=66328). The array is also known as Khayyam's triangle, Yang Hui's triangle, or Tartaglia's triangle, reflecting independent traditions of study across several mathematical cultures[1](https://courses.maths.ox.ac.uk/mod/page/view.php?id=66328). The recurrence also serves computation: building the triangle row by row computes every \( \binom{n}{k} \) for \( k = 0, \dots, n \) in \( O(n^{2}) \) time, and avoids the arithmetic overflow that direct evaluation from factorials can cause even when the final coefficient fits in an integer[5](https://eecs.umich.edu/courses/eecs380/ALG/binom.html) • [6](https://www3.cs.stonybrook.edu/~skiena/392/lectures/week6/).
## Proof by induction
The standard proof proceeds by induction on \( n \)[1](https://courses.maths.ox.ac.uk/mod/page/view.php?id=66328). The base case \( n = 0 \) holds because both sides equal 1. Assuming the formula for \( n \), one writes
\[ (x+y)^{n+1} = (x+y) \left( \sum_{k=0}^{n} \binom{n}{k} x^{k} y^{n-k} \right) \]
and expands the product. Collecting terms with the same total degree and applying Pascal's rule \( \binom{n}{k-1} + \binom{n}{k} = \binom{n+1}{k} \) produces exactly the coefficients required for \( n+1 \)[1](https://courses.maths.ox.ac.uk/mod/page/view.php?id=66328). The same induction structure appears in elementary treatments, where each expansion of \( (a+b)^{k+1} \) is obtained by multiplying the known expansion of \( (a+b)^{k} \) by \( (a+b) \)[7](https://math.libretexts.org/Courses/Cosumnes_River_College/Math_372%3A_College_Algebra_for_Calculus/07%3A_Sequences_and_Series_Mathematical_Induction_and_the_Binomial_Theorem/7.04%3A_The_Binomial_Theorem).
## History
**Medieval Islamic mathematics.** The theorem is presumed to have been discovered by al-Karaji (ca. 1029) and was used by several subsequent mathematicians; a work of al-Zanjani (d. 1262) contains an example illustrating the expansion up to the seventh power of the sum of two quantities, which points to the theorem being a widespread concept in medieval Islamic mathematics[2](https://www.sciencedirect.com/science/article/pii/031508608090004X). In *Al-Fakhri*, al-Karaji computed \( (a+b)^{3} \), and in *Al-Badi* he computed \( (a-b)^{3} \) and \( (a+b)^{4} \); the general construction of the Pascal triangle is described in the later writings of his successor al-Samawal, who preserved a step-by-step account of building each column of coefficients from the previous one[8](https://mathshistory.st-andrews.ac.uk/Biographies/Al-Karaji/). Al-Karaji also used a form of mathematical induction in this work, arguing case by case up to about \( n = 5 \) before remarking that the process continues indefinitely[8](https://mathshistory.st-andrews.ac.uk/Biographies/Al-Karaji/).
**Pascal and Newton.** Blaise Pascal was not the first to study the triangle, but his *Treatise on the Arithmetical Triangle* was the most important work on the topic, and, through the work of John Wallis, Pascal's treatment of the binomial coefficients led [Isaac Newton](https://www.edgepedia.org/isaac-newton) to his discovery of the general binomial theorem for fractional and negative powers[9](https://mathshistory.st-andrews.ac.uk/Biographies/Pascal/). In the paper *De Analysi per aequationes numero terminorum infinitas*, Newton showed how to calculate the coefficients in binomial formulas directly, rather than line by line through Pascal's triangle[10](https://mathshistory.st-andrews.ac.uk/Strick/newton.pdf). In modern notation his series reads
\[ (1+x)^{n} = 1 + n \cdot x + \frac{n(n-1)}{2} \cdot x^{2} + \frac{n(n-1)(n-2)}{3 \cdot 2} \cdot x^{3} + \cdots, \]
which terminates when \( n \) is a nonnegative integer and continues indefinitely for negative and fractional exponents; for example \( (1+x)^{-3} = 1 - 3x + 6x^{2} - 10x^{3} + 15x^{4} - \cdots \)[10](https://mathshistory.st-andrews.ac.uk/Strick/newton.pdf). Newton checked such expansions by multiplying or squaring the series and observing that all terms with positive exponents cancel, leaving 1; he also used the square-root series to approximate roots, estimating \( \sqrt{3} \approx 1{,}734 \) by writing \( \sqrt{3} = 2\sqrt{1 - 1/4} \)[10](https://mathshistory.st-andrews.ac.uk/Strick/newton.pdf). Later writers noted that expanding radicals into infinite series by means of the binomial theorem, which no one had done before, was central to [Newton's method](https://www.edgepedia.org/newtons-method) for squaring curves[11](https://mathshistory.st-andrews.ac.uk/Extras/Bossut_Chapter_V/).
## The generalized binomial theorem and the binomial series
The falling-product expression \( \binom{r}{k} = r(r-1)(r-2) \cdots (r-k+1)/k! \) makes sense for any real \( r \) as long as \( k \) is a nonnegative integer, and these generalized coefficients retain Pascal's rule \( \binom{r}{k} = \binom{r-1}{k-1} + \binom{r-1}{k} \)[3](https://www.whitman.edu/mathematics/cgt_online/book/section03.01.html). With this definition, Newton's binomial theorem states that for any real \( r \) that is not a nonnegative integer,
\[ (1+x)^{r} = \sum_{i=0}^{\infty} \binom{r}{i} x^{i} \qquad \text{for } -1 < x < 1. \]
The series is the Maclaurin series of \( (1+x)^{r} \); showing that it actually equals \( (1+x)^{r} \) requires more work and is treated in introductory real analysis texts[3](https://www.whitman.edu/mathematics/cgt_online/book/section03.01.html).
The complex-variable form is called the binomial series. For any complex \( \alpha \) that is not a nonnegative integer, the series \( \sum_{n=0}^{\infty} \binom{\alpha}{n} z^{n} \) has radius of convergence \( R = 1 \): it converges absolutely for \( |z| < 1 \) and diverges for \( |z| > 1 \). On the circle \( |z| = 1 \) the behavior depends on \( \operatorname{Re}(\alpha) \): the series converges absolutely when \( \operatorname{Re}(\alpha) > 0 \), diverges at \( z = -1 \) when \( -1 < \operatorname{Re}(\alpha) \leqslant 0 \), and diverges everywhere on the circle when \( \operatorname{Re}(\alpha) \leqslant -1 \). Wherever it converges, its sum is the principal value of \( (1+z)^{\alpha} \)[12](https://www.math.ualberta.ca/~isaac/math311/s14/binomial_series.pdf).
A useful negative-index identity follows from the generalized definition: \( \binom{-n}{r} = (-1)^{r} \binom{n+r-1}{r} \), which for example gives \( (1-x)^{-1} = 1 + x + x^{2} + \cdots \)[13](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_%28Morris%29/02%253A_Enumeration/07%253A_Generating_Functions/7.02%253A_The_Generalized_Binomial_Theorem). The generalized theorem is also the source of standard expansions such as those of \( e^{x} \), \( a^{x} \), and \( \log(1+x) \) in elementary courses, and of approximation techniques[14](https://nepalenotes.com/2022/06/class-12-mathematics-binomial-theorem/).
## Applications
**Probability.** In a sequence of \( N \) independent trials with two outcomes \( A \) and \( B \) of probabilities \( p_A \) and \( p_B = 1 - p_A \), the probability that \( A \) occurs exactly \( N_A \) times is
\[ p_N(N_A) = \frac{N!}{N_A!(N - N_A)!} \, p_A^{N_A} p_B^{N - N_A}. \]
The prefactor is precisely the coefficient obtained in the binomial expansion of \( (p_A + p_B)^N \), and counts the possible orderings of the \( N_A \) events \( A \) and \( N_B = N - N_A \) events \( B \); the probability generating function of the distribution is \( (p_A e^{\lambda} + p_B)^{N} \)[4](https://web.mit.edu/8.Math/www/lectures/lec15/4.1.3.pdf).
**Coefficient identities.** Substituting particular values into the theorem evaluates sums of binomial coefficients: setting \( x = y = 1 \) gives \( 2^{n} = \sum_{k} \binom{n}{k} \), the row sum of Pascal's triangle, and setting \( x = 1 \), \( y = -1 \) gives the alternating sum \( \sum_{k} (-1)^{k} \binom{n}{k} = 0 \)[15](https://neelmishra.github.io/blog/cp/combinatorics-1/binomial-coefficients.html). The [National Institute of Standards and Technology](https://www.edgepedia.org/national-institute-of-standards-and-technology)'s *Digital Library of Mathematical Functions* records further identities, including the hockey-stick sum \( \binom{m+1}{n+1} = \sum_{k=n}^{m} \binom{k}{n} \) and the asymptotic size of the central coefficient, \( \binom{2n}{n} \sim 4^{n}/\sqrt{\pi n} \) as \( n \to \infty \)[16](https://dlmf.nist.gov/26.3).
**Generalizations.** The multinomial theorem extends the expansion to sums of more than two terms, with coefficients \( N!/(N_A! N_B! \cdots N_M!) \)[4](https://web.mit.edu/8.Math/www/lectures/lec15/4.1.3.pdf). When the variables do not commute, the ordinary theorem fails, but a \( q \)-analog holds: if \( yx = qxy \) for a scalar \( q \) commuting with both variables, then \( (x+y)^{n} = \sum_{j=0}^{n} \begin{bmatrix} n \\ j \end{bmatrix} x^{j} y^{n-j} \), where \( \begin{bmatrix} n \\ j \end{bmatrix} \) is the Gaussian \( q \)-binomial coefficient[17](https://lib.ysu.am/open_books/418595.pdf). Such noncommutative binomial formulas are used in quantum group theory and noncommutative geometry[18](https://www.sciencedirect.com/science/article/abs/pii/S0393044099000327), and the quantum binomial theorem has been applied to refine the ordinary binomial distribution by encoding the sequence of successes and failures in the exponent of \( q \)[19](https://arxiv.org/abs/2009.12641).
## References
1. [Introduction to University Mathematics (2026-27): The natural numbers and induction, Mathematical Institute, University of Oxford](https://courses.maths.ox.ac.uk/mod/page/view.php?id=66328)
2. [Mohammad Yadegari, "The binomial theorem: A widespread concept in medieval Islamic mathematics", Historia Mathematica 7(4), 401-406 (1980)](https://www.sciencedirect.com/science/article/pii/031508608090004X)
3. ["3.1 Newton's Binomial Theorem", Whitman College, Calculus of Generating Functions](https://www.whitman.edu/mathematics/cgt_online/book/section03.01.html)
4. [MIT 8.Math lecture notes, section 4.1.3, Binomial distribution](https://web.mit.edu/8.Math/www/lectures/lec15/4.1.3.pdf)
5. [Data Structures and Algorithms: Binomial Coefficients, University of Michigan EECS](https://eecs.umich.edu/courses/eecs380/ALG/binom.html)
6. [CSE 392 Combinatorics, Week 6, Stony Brook University](https://www3.cs.stonybrook.edu/~skiena/392/lectures/week6/)
7. ["7.4: The Binomial Theorem", Mathematics LibreTexts](https://math.libretexts.org/Courses/Cosumnes_River_College/Math_372%3A_College_Algebra_for_Calculus/07%3A_Sequences_and_Series_Mathematical_Induction_and_the_Binomial_Theorem/7.04%3A_The_Binomial_Theorem)
8. ["Al-Karaji (953 - 1029)", MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Al-Karaji/)
9. ["Blaise Pascal (1623 - 1662)", MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Pascal/)
10. [Heinz Klaus Strick, "Isaac Newton", MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Strick/newton.pdf)
11. [Charles Bossut, A General History of Mathematics, Chapter V, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Extras/Bossut_Chapter_V/)
12. ["The Binomial Series", Math 311, University of Alberta](https://www.math.ualberta.ca/~isaac/math311/s14/binomial_series.pdf)
13. ["7.2: The Generalized Binomial Theorem", Mathematics LibreTexts](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_%28Morris%29/02%253A_Enumeration/07%253A_Generating_Functions/7.02%253A_The_Generalized_Binomial_Theorem)
14. [Class 12 Mathematics Binomial Theorem Notes, Nepal Enotes](https://nepalenotes.com/2022/06/class-12-mathematics-binomial-theorem/)
15. [Neel Mishra, "Binomial Coefficients: Precomputation & Modular Arithmetic"](https://neelmishra.github.io/blog/cp/combinatorics-1/binomial-coefficients.html)
16. [DLMF §26.3, Lattice Paths: Binomial Coefficients, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/26.3)
17. [Kac and Cheung, Quantum Calculus, Universitext, chapter 5](https://lib.ysu.am/open_books/418595.pdf)
18. ["A non-commutative binomial formula", Journal of Geometry and Physics 32(4), 349-363 (2000)](https://www.sciencedirect.com/science/article/abs/pii/S0393044099000327)
19. [Andrew V. Sills, "A refinement of the binomial distribution using the quantum binomial theorem", arXiv:2009.12641](https://arxiv.org/abs/2009.12641)
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