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Binomial Coefficient



Definition
Binomial Coefficient Notation
Identities
Pascal's Triangle
The Multinomial Coefficient
Related Concepts
Binomial Coefficient at a Glance
Binomial Coefficient FAQ



The Object Behind Combinations


The combination formula (nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!} first appears as the answer to a counting question: how many kk-element subsets does an nn-element set have? But the same expression turns up in far more than subset-counting problems. It organizes polynomial expansions, governs probability distributions, encodes symmetric structures, and satisfies a network of identities that have nothing obvious to do with subsets.

Treating (nk)\binom{n}{k} as a stand-alone mathematical object — independent of any specific counting scenario — reveals structure that the counting interpretation alone does not expose. This page covers the formal definition and notation, the algebraic identities the coefficient satisfies, the visual organization of all binomial coefficients into Pascal's triangle, and the generalization to multinomial coefficients.



Definition


For non-negative integers nn and kk with 0≤k≤n0 \le k \le n, the binomial coefficient is

Binomial Coefficient
(nk)=n!k! (n−k)!.\binom{n}{k} = \frac{n!}{k!\,(n-k)!}.
Learn more about this formula: Binomial Coefficient →


Boundary cases follow directly: (n0)=(nn)=1\binom{n}{0} = \binom{n}{n} = 1 for every n≥0n \ge 0, and (nk)=0\binom{n}{k} = 0 whenever k>nk > n.

Generalization to Real Upper Index


The definition extends beyond non-negative integers by rewriting the numerator as a descending product:

Generalized Binomial Coefficient
(xk)=x(x−1)(x−2)⋯(x−k+1)k!,k∈Z≥0,  x∈R (or C).\binom{x}{k} = \frac{x(x-1)(x-2) \cdots (x-k+1)}{k!}, \quad k \in \mathbb{Z}_{\ge 0}, \; x \in \mathbb{R} \text{ (or } \mathbb{C}\text{)}.
Learn more about this formula: Generalized Binomial Coefficient →


Here xx can be any real or complex number and kk remains a non-negative integer. The numerator is a polynomial of degree kk in xx, so (xk)\binom{x}{k} is itself a polynomial in xx. This is the form that appears in Newton's generalized binomial theorem, where the upper index is no longer required to be a non-negative integer.

The combinatorial interpretation — the number of combinations of kk items chosen from nn — remains the most intuitive entry point, but the algebraic definition is what extends to non-integer settings.

Binomial Coefficient Notation

Notation

Binomial Coefficient Notation

One object, four spellings, and a stack that is not a fraction. The read-aloud form — "nn choose kk" — is the key that unlocks all of them; in LaTeX the stacked form is \binom{n}{k}.
n!n! inside every formula is owned by permutation notation; the multinomial extension (nk1,…,kr)\binom{n}{k_1, \ldots, k_r} is built in The Multinomial Coefficient below.
(nk)\binom{n}{k}
n choose k
The standard modern form: two numbers stacked in parentheses, no bar between them — the top chooses from, the bottom chooses how many. Reading it as a fraction is the founding error; (62)=15\binom{6}{2} = 15, not 33.
CasesThe same mark survives the definition's stretch: (xk)\binom{x}{k} with real or complex upper index — Generalization to Real Upper Index above — and grows extra lower entries in the multinomial form of The Multinomial Coefficient below.
Same glyph elsewhereA two-entry column vector wears identical clothing — two stacked numbers in parentheses; context (counting versus coordinates) is the only separator.
C(n,k)C(n, k) · nCk_nC_k
C n k; n choose k on a calculator
The linear spellings: C(n,k)C(n, k) in introductory texts — function notation borrowed for a count — and nCk_nC_k on calculator keys and displays, the combinatorial sibling of nPr_nP_r from permutation notation.
CasesAll spellings agree with the stacked form: C(n,k)=nCk=(nk)C(n, k) = {}_nC_k = \binom{n}{k}; the linear forms survive where stacking is impossible — plain-text input, spreadsheet formulas, calculator screens.
Do not confuseA function named C. C(n,k)C(n, k) looks like an evaluation of some function CC — harmless until CC is also a constant or a circle in the same problem; the stacked form dodges the letter entirely.
CnkC_n^k vs CknC_k^n
C with n and k in the corners — order varies
The European corner forms: French and Russian traditions attach nn and kk as subscript and superscript — but which corner holds which number varies by country and by book. The mark alone does not say.
CasesWhere the convention is fixed it is fixed hard — Russian texts write CnkC_n^k with the pool nn below; some French sources invert. Checking against a known value (C52=10C_5^2 = 10 or C25C_2^5?) settles any book in one line.
Do not confuseA power. CnkC_n^k reads dangerously like "CnC_n raised to the kk" — the superscript is an index, not an exponent; the same crowded-corner hazard the inverse marks live with.

Identities


The binomial coefficient satisfies a network of identities, many of which are not visible from the counting interpretation alone.

Symmetry


Binomial Symmetry Identity
(nk)=(nn−k).\binom{n}{k} = \binom{n}{n-k}.
Learn more about this formula: Binomial Symmetry Identity →


Choosing which kk items to include is equivalent to choosing which n−kn-k items to exclude.

Pascal's Rule


Pascal's Rule
(nk)=(n−1k−1)+(n−1k).\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}.
Learn more about this formula: Pascal's Rule →


Fix a particular element of the nn-set. A kk-subset either contains that element — in which case the remaining k−1k-1 items come from the other n−1n-1 — or does not, in which case all kk items come from the other n−1n-1. The two cases partition the kk-subsets, and adding their counts gives the identity. This recursion is the foundation of Pascal's triangle.

Absorption Identity


Absorption Identity
k(nk)=n(n−1k−1).k \binom{n}{k} = n \binom{n-1}{k-1}.
Learn more about this formula: Absorption Identity →


Both sides count the number of ways to choose a kk-element subset and then designate one of its members as distinguished.

Row Sum


Binomial Row Sum
∑k=0n(nk)=2n.\sum_{k=0}^{n} \binom{n}{k} = 2^n.
Learn more about this formula: Binomial Row Sum →


The total number of subsets of an nn-set, summed over all possible sizes, equals the size of the power set.

Alternating Row Sum


Alternating Binomial Sum
∑k=0n(−1)k(nk)=0,n≥1.\sum_{k=0}^{n} (-1)^k \binom{n}{k} = 0, \quad n \ge 1.
Learn more about this formula: Alternating Binomial Sum →


The number of even-sized subsets of an nn-set equals the number of odd-sized subsets.

Vandermonde's Identity


Vandermonde's Identity
(m+nr)=∑k=0r(mk)(nr−k).\binom{m+n}{r} = \sum_{k=0}^{r} \binom{m}{k} \binom{n}{r-k}.
Learn more about this formula: Vandermonde's Identity →


Choosing rr items from the disjoint union of an mm-set and an nn-set partitions according to how many of the chosen items come from each side.

Hockey Stick Identity


Hockey Stick Identity
∑i=kn(ik)=(n+1k+1).\sum_{i=k}^{n} \binom{i}{k} = \binom{n+1}{k+1}.
Learn more about this formula: Hockey Stick Identity →


A sum running down a diagonal of Pascal's triangle equals a single entry one row below.

Most of these identities admit both algebraic and combinatorial proofs. The combinatorial proofs typically use double counting: the same set is enumerated by two different strategies, and the two expressions for its size are equated.
Identity Formula Combinatorial meaning
Symmetry C(n, k) = C(n, n − k) choosing the k included items ↔ choosing the n − k excluded items
Pascal's rule C(n, k) = C(n−1, k−1) + C(n−1, k) a fixed element is either in the subset or out of it
Absorption k · C(n, k) = n · C(n−1, k−1) choose a k-subset, then distinguish one of its members
Row sum Σk=0n C(n, k) = 2n total number of subsets of an n-set
Alternating row sum Σk=0n (−1)k C(n, k) = 0 for n ≥ 1 even-sized subsets and odd-sized subsets are equally numerous
Vandermonde's identity C(m+n, r) = Σk=0r C(m, k) C(n, r−k) partition the chosen items between the two source sets
Hockey stick Σi=kn C(i, k) = C(n+1, k+1) a diagonal sum in Pascal's triangle equals a single entry one row below

Pascal's Triangle


Pascal's rule arranges all binomial coefficients into a triangular array. Row nn, indexed starting from row 00, contains the n+1n+1 values

(n0),(n1),…,(nn).\binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}.


Each interior entry equals the sum of the two entries directly above it — the visual form of Pascal's rule.

What the Triangle Encodes


The symmetry identity (nk)=(nn−k)\binom{n}{k} = \binom{n}{n-k} is immediate from the array: each row reads the same forwards and backwards.

The diagonals of the triangle hold familiar sequences:

• The first diagonal contains the constants 1,1,1,…1, 1, 1, \ldots
• The second contains the natural numbers 1,2,3,4,…1, 2, 3, 4, \ldots
• The third contains the triangular numbers 1,3,6,10,…1, 3, 6, 10, \ldots
• The fourth contains the tetrahedral numbers 1,4,10,20,…1, 4, 10, 20, \ldots

Sums along the shallow diagonals — diagonals that move one step down and one step left at each entry — produce the Fibonacci numbers.

The row sums double at each step, since ∑k(nk)=2n\sum_k \binom{n}{k} = 2^n.

Computing Coefficients


For computing individual binomial coefficients, Pascal's triangle is the most efficient route when the values needed are small. Beyond moderate size, the explicit formula or the absorption identity is faster.
Diagonal Sequence name First few values Closed form (n-th)
1st (k = 0) constants 1, 1, 1, 1, … 1
2nd (k = 1) natural numbers 1, 2, 3, 4, … n
3rd (k = 2) triangular numbers 1, 3, 6, 10, 15, … n(n+1) ⁄ 2
4th (k = 3) tetrahedral numbers 1, 4, 10, 20, 35, … n(n+1)(n+2) ⁄ 6

The Multinomial Coefficient


The binomial coefficient counts ways to split nn items into two groups of sizes kk and n−kn-k. The multinomial coefficient is the natural generalization to more than two groups.

Definition


For non-negative integers k1,k2,…,krk_1, k_2, \ldots, k_r with k1+k2+⋯+kr=nk_1 + k_2 + \cdots + k_r = n,

Multinomial Coefficient
(nk1,k2,…,kr)=n!k1! k2!⋯kr!,k1+k2+⋯+kr=n.\binom{n}{k_1, k_2, \ldots, k_r} = \frac{n!}{k_1! \, k_2! \cdots k_r!}, \quad k_1 + k_2 + \cdots + k_r = n.
Learn more about this formula: Multinomial Coefficient →


The binomial coefficient is the case r=2r = 2:

(nk)=(nk, n−k).\binom{n}{k} = \binom{n}{k, \, n-k}.


Combinatorial Interpretation


The multinomial coefficient counts the number of distinct arrangements of nn items in which k1k_1 are of type 1, k2k_2 are of type 2, and so on through krk_r items of type rr. This is the same count that appears in permutations with identical items — the two viewpoints (partitioning into groups versus arranging with repetition of types) refer to the same enumeration.

The multinomial coefficient is also the natural coefficient in the multinomial theorem, which expands (x1+x2+⋯+xr)n(x_1 + x_2 + \cdots + x_r)^n.

Related Concepts


• Probability — the binomial distribution and several related discrete distributions are built directly on the binomial coefficient.

• Algebra — polynomial identities, generating functions, and finite difference calculus all rely on binomial expansions and binomial-coefficient identities.

Binomial Coefficient at a Glance


The page covered the formal definition and notation, the network of identities the coefficient satisfies, the Pascal's-triangle organization, and the multinomial generalization. The table below collects the formula, boundary values, and generalizations in one reference card.
Concept Statement Example
Standard formula C(n, k) = n! ⁄ (k! · (n − k)!) for 0 ≤ k ≤ n C(5, 2) = 120 ⁄ (2 · 6) = 10
Combinatorial meaning number of k-element subsets of an n-element set 2-subsets of {a, b, c, d, e}: 10
Read aloud "n choose k" C(5, 2) reads "5 choose 2"
Edge values C(n, 0) = C(n, n) = 1 for every n ≥ 0 C(7, 0) = C(7, 7) = 1
Linear values C(n, 1) = C(n, n − 1) = n C(7, 1) = 7
Out of range C(n, k) = 0 whenever k > n or k < 0 C(3, 5) = 0
Real upper index C(x, k) = x(x−1)(x−2)…(x−k+1) ⁄ k! — a polynomial of degree k in x C(½, 2) = (½ · −½) ⁄ 2 = −1⁄8
Pascal's-triangle row row n holds C(n, 0), C(n, 1), …, C(n, n) row 4: 1, 4, 6, 4, 1
Multinomial generalization C(n; k₁, …, kr) = n! ⁄ (k₁! · k₂! · … · kr!) with k₁+…+kr = n C(6; 2, 2, 2) = 720 ⁄ 8 = 90
Where it appears subset counts, polynomial expansions (binomial theorem), discrete probability (binomial distribution), finite-difference calculus (x + y)n coefficients are C(n, k)

Binomial Coefficient FAQ

How do you read the notation n choose k?

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As the number of ways to select k items from n when order does not matter. The stacked pair in brackets carries no fraction bar, despite looking like one, and is spoken as n choose k rather than n over k. Writing it as a fraction is a frequent transcription error.Read more →

Why does choosing k items equal choosing n minus k?

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Because selecting which items to take is the same decision as selecting which to leave behind. Every choice of k determines a unique choice of the remaining items, so the two counts match exactly. This symmetry halves the work when computing values near the middle of a row of Pascal's triangle.Read more →

What is Pascal's rule?

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It says each binomial coefficient is the sum of the two directly above it. The reason is a case split: either a particular item is included in the selection or it is not, and counting both cases separately gives the two terms. This rule is what builds Pascal's triangle row by row.Read more →

Why is zero factorial equal to one?

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Because there is exactly one way to arrange nothing: the empty arrangement. Defining it as one also keeps the formulas consistent, since choosing all n items or choosing none should both give a count of one, and only this convention makes the binomial coefficient formula produce that.Read more →