Binomial

The binomial coefficient $\binom{n}{k}$, the number of k-element subsets of an n-element set.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaBinomial coefficient↗MathWorldBinomialCoefficient↗WikidataQ209875↗Encyclopedia of MathematicsBinomial_coefficients↗Britannicatopic/binomial-coefficient↗DLMF26.3↗1.2.i↗FungrimBinomial✓↗Fungrim entryOEISA073354↗Wolfram LanguageBinomial!↗

Binomial(n, k)the binomial coefficient $\binom{n}{k}$.

SageMathenumeratio_binomial(n, k)SymPybinomial(n, k)↗mpmathbinomial(n, k)Oscarbinomial(ZZ(n), ZZ(k))Juliabinomial(ZZ(n), ZZ(k))Rustbinomial(n, k)

Typed signatures

Type	Package	Applies
(complex | infinity, complex | infinity) -> number	analyticreplaces compute-engine	always
(complex | infinity, complex | infinity) -> number	compute-engine	always
(complex | infinity, complex | infinity) -> number	number-theoryreplaces analytic	always

Domain: Combinatorics

Details

• The general definition runs through the Gamma function: $\binom{n}{k} = \frac{\Gamma(n+1)}{\Gamma(k+1)\,\Gamma(n-k+1)}$.
• Symmetric in its arguments: $\binom{n}{k} = \binom{n}{n-k}$.
• Pascal's rule builds each row from the last: $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$.
• Row sums give $\sum_{k=0}^{n} \binom{n}{k} = 2^n$.
• compute-engine evaluates only integer n and k: k outside $[0, n]$ gives 0 for nonnegative n, while a negative n switches to the generalized falling-factorial formula rather than the Gamma form.
• A Gaussian or otherwise complex n and/or k also evaluates, through the same Gamma-function identity: exact for an integer k, numeric otherwise.

Plot

ComplexPlot

real slice

Examples

16 more kept as data, checked against the oracles

Basic #

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In$\binom{5}{2}$Out$10$

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In$\binom{10}{3}$Out$120$

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In$\binom{4}{2}$Out$6$

Choosing none of n is always 1 way

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In$\binom{5}{0}$Out$1$

Choosing all of n is always 1 way

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In$\binom{5}{5}$Out$1$

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In$\binom{8}{4}$Out$70$

Symbolic n: $\binom{n}{0} = 1$ for any n

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In$\binom{n}{0}$Out$1$

Symbolic n: $\binom{n}{1} = n$

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In$\binom{n}{1}$Out$n$

$\binom{n}{n-1} = n$ for symbolic n

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In$\binom{n}{n-1}$Out$n$

Scope #

Threads element-wise over a list, as Wolfram's Listable heads do

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In$\binom{\bigl\lbrack2, 3, 5, 7, 11\bigr\rbrack}{3}$Out$\bigl\lbrack0, 1, 10, 35, 165\bigr\rbrack$

Half-integer arguments evaluate exactly through the Gamma function: here $n - k = 1$, so the answer is just $n = \frac{9}{2}$

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In$\binom{\frac{9}{2}}{\frac{7}{2}}$Out$\frac{9}{2}$

Real arguments evaluate through the Gamma function

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In$\binom{8.5}{-4.2}$Out$0.000\,060\,499\,248\,402\,283\,480\,067\,6$

A real n with integer k: $\frac{0.5\,(-0.5)(-1.5)}{3!} = 0.0625$, up to the last bignum digit

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In$\binom{0.5}{3}$Out$0.062\,500\,000\,000\,000\,000\,000\,1$

A rational $n$ with integer $k$ gives the exact falling-factorial value $\binom{1/2}{2} = -\frac{1}{8}$

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In$\binom{\frac{1}{2}}{2}$Out$-\frac{1}{8}$

$\binom{-1/2}{3} = -\frac{5}{16}$, the central-binomial series coefficient

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In$\binom{-\frac{1}{2}}{3}$Out$-\frac{5}{16}$

$\binom{n}{n} = 1$ for symbolic n

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In$\binom{n}{n}$Out$1$

A Gaussian-integer n: $\binom{1+i}{5} = -\frac{1+i}{12}$, exact

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In$\binom{1+i}{5}$Out$-\frac{1}{12}-\frac{1}{12}i$

Complex n and k evaluate through the Gamma function

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In$\binom{2+i}{7-3i}$Out$-75.468\,347\,382\,223\,04+106.815\,265\,970\,790\,55i$

Interval arithmetic in $k$: $\binom{1/2}{k}$ is decreasing here, so the endpoints swap

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In$\binom{\frac{1}{2}}{\mathrm{Interval}(0.5, 0.6)}$Out$\mathrm{Interval}(0.928\,145\,553\,850\,705\,4, 1)$

Non-integer arguments are fine, evaluated to 20 digits

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In$N(\binom{\frac{7}{3}}{\frac{1}{5}}, 20)$Out$1.333\,125\,424\,465\,028\,652\,2$

A symbolic top with integer bottom expands to a polynomial in $n$

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In$\binom{n}{4}$Out$\frac{1}{24}n×(n-3)(n-2)(n-1)$

Applied to a matrix, a polynomial in the matrix with exact entries

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\binom{\operatorname{\_1}}{2}, \bigl\lbrack\bigl\lbrack\frac{5π}{6}, 0\bigr\rbrack, \bigl\lbrack\frac{3π}{2}, -\frac{π}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{1}{72}×5π(5π-6), 0\bigr\rbrack, \bigl\lbrack\frac{1}{64}×5π(5π-6)-\frac{1}{64}×9π(2+π), \frac{π(2+π)}{8}\bigr\rbrack\bigr\rbrack$

Applications #

The number of distinct 5-card hands dealt from a standard 52-card deck

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In$\binom{52}{5}$Out$2\,598\,960$

The probability of exactly 3 heads in 10 fair coin tosses

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In$\frac{\binom{10}{3}}{2^{10}}$Out$\frac{15}{128}$

Properties #

Symmetry: $\binom{n}{k} = \binom{n}{n-k}$

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In$\binom{10}{3}=\binom{10}{7}$Out$\mathrm{True}$

Pascal's rule: $\binom{n-1}{k-1} + \binom{n-1}{k} = \binom{n}{k}$

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In$\binom{9}{2}+\binom{9}{3}=\binom{10}{3}$Out$\mathrm{True}$

Row sums give $\sum_k \binom{n}{k} = 2^n$, here at $n = 4$

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In$\binom{4}{0}+\binom{4}{1}+\binom{4}{2}+\binom{4}{3}+\binom{4}{4}=2^4$Out$\mathrm{True}$

Sum of squares along a row: $\sum_k \binom{n}{k}^2 = \binom{2n}{n}$

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In$\sum_{k=0}^6\binom{6}{k}^2=\binom{12}{6}$Out$\mathrm{True}$

Vandermonde's identity: $\sum_k \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}$

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In$\sum_{k=0}^3\binom{5}{k}\binom{4}{3-k}=\binom{9}{3}$Out$\mathrm{True}$

The alternating row sum vanishes: $\sum_k (-1)^k \binom{n}{k} = 0$ for $n > 0$

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In$\sum_{k=0}^6(-1)^{k}\binom{6}{k}$Out$0$

The factorial form $\binom{n}{k} = \frac{n!}{k!\,(n-k)!}$. See Factorial

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In$\binom{7}{3}=\frac{7!}{3!×4!}$Out$\mathrm{True}$

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In$\binom{\frac{1}{2}}{-\frac{1}{2}}+\binom{\frac{1}{2}}{\frac{1}{2}}$Out$\frac{3}{2}$

Known value · Wolfram MathematicalFunctionData, Binomial, Pascal's rule

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In$\sum_{k=0}^2\binom{2}{k}^2\binom{k+2}{k}^2$Out$73$

Known value · Wolfram MathematicalFunctionData, Binomial, Apéry's numbers

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In$\sum_{k=0}^{\lfloor\frac{1}{2}\rfloor}\binom{2}{2k+1}×2^{k}$Out$2$

Known value · Wolfram MathematicalFunctionData, Binomial, Pell numbers

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In$\binom{\frac{1}{2}}{\frac{1}{2}}+\binom{\frac{1}{2}}{\frac{3}{2}}$Out$1$

Known value · Wolfram MathematicalFunctionData, Binomial, FunctionalEquations

Possible issues #

Choosing more elements than available ($k > n$) gives 0, not an error

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In$\binom{5}{7}$Out$0$

A negative k also gives 0

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In$\binom{5}{-1}$Out$0$

A negative n uses the generalized formula $\binom{n}{k} = \frac{n(n-1)\cdots(n-k+1)}{k!}$ rather than 0

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In$\binom{-5}{2}$Out$15$

Both arguments negative integers: the limiting definition gives $\binom{-3}{-5} = \binom{-3}{2} = 6$, not 0

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In$\binom{-3}{-5}$Out$6$

Neat examples #

The central binomial coefficient $\binom{100}{50}$ has 30 digits

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In$\binom{100}{50}$Out$100\,891\,344\,545\,564\,193\,334\,812\,497\,256$

About

Examples	56
Checked against	Wolfram Language 55 agree1 differSageMath 51 agree3 differSymPy 49 agree5 differ2 unevaluatedmpmath 41 agree1 differmathlib4 16 agree19 errorOscar 26 agree2 differ8 errorJulia 27 agree3 differ12 errorRust 17 agree13 differ10 error
Crosswalk	WikipediaMathWorldWikidataEncyclopedia of MathematicsBritannicaDLMF ×2FungrimFungrim entry ×23OEISWolfram LanguageSageMathSymPympmathOscarJuliaRust

Implementation

mappedwolframpackages/oracle/src/mappings.ts

Wolfram's (and compute-engine's) Binomial extends to negative n and k via the reflection identities in its docs (e.g. Binomial[5,-2] = 0, Binomial[-7,2] = 28, Binomial[-5,-7] = 15); the crates behind sage/oscar/julia/rust bottom out at unsigned or non-negative-only integer types and diverge there — a convention gap, not a bug on either side. Verified against wolframscript.

mappedsympypackages/oracle/src/mappings.ts

Wolfram's (and compute-engine's) Binomial extends to negative n and k via the reflection identities in its docs (e.g. Binomial[5,-2] = 0, Binomial[-7,2] = 28, Binomial[-5,-7] = 15); the crates behind sage/oscar/julia/rust bottom out at unsigned or non-negative-only integer types and diverge there — a convention gap, not a bug on either side. Verified against wolframscript.

mappedsagepackages/oracle/src/mappings.ts

Sage's binomial needs m or x-m an integer; enumeratio_binomial falls back to the Gamma ratio otherwise (run.ts) and answers a complex infinity at a pole of Gamma(x+1), as ours does. A negative n with an integer k agrees (binomial(-5, 2) = 15); two negative integers give 0 where Wolfram and compute-engine take the limit (Binomial[-2,-2] = 1), a convention gap, not a bug on either side.

mappedoscarpackages/oracle/src/mappings.ts

Wolfram's (and compute-engine's) Binomial extends to negative n and k via the reflection identities in its docs (e.g. Binomial[5,-2] = 0, Binomial[-7,2] = 28, Binomial[-5,-7] = 15); the crates behind sage/oscar/julia/rust bottom out at unsigned or non-negative-only integer types and diverge there — a convention gap, not a bug on either side. Verified against wolframscript.

mappedjuliapackages/oracle/src/mappings.ts

Wolfram's (and compute-engine's) Binomial extends to negative n and k via the reflection identities in its docs (e.g. Binomial[5,-2] = 0, Binomial[-7,2] = 28, Binomial[-5,-7] = 15); the crates behind sage/oscar/julia/rust bottom out at unsigned or non-negative-only integer types and diverge there — a convention gap, not a bug on either side. Verified against wolframscript.

mappedmathlib4packages/oracle/src/mappings.ts

Wolfram's (and compute-engine's) Binomial extends to negative n and k via the reflection identities in its docs (e.g. Binomial[5,-2] = 0, Binomial[-7,2] = 28, Binomial[-5,-7] = 15); the crates behind sage/oscar/julia/rust bottom out at unsigned or non-negative-only integer types and diverge there — a convention gap, not a bug on either side. Verified against wolframscript.

mappedrustpackages/oracle/src/mappings.ts

Wolfram's (and compute-engine's) Binomial extends to negative n and k via the reflection identities in its docs (e.g. Binomial[5,-2] = 0, Binomial[-7,2] = 28, Binomial[-5,-7] = 15); the crates behind sage/oscar/julia/rust bottom out at unsigned or non-negative-only integer types and diverge there — a convention gap, not a bug on either side. Verified against wolframscript.

mappedmpmathpackages/oracle/src/mappings.ts

See also: Factorial, Multinomial, Pochhammer