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Binomial Coefficient



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



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 and Notation


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. The expression is read aloud as "nn choose kk".

Notation Variants


Several notations are in use for the same object:

• (nk)\binom{n}{k} — the standard modern form, used throughout mathematics
• C(n,k)C(n,k) — common in introductory texts
• CnkC_n^k or CknC_k^n — used in some European traditions; the position of the subscript and superscript varies by source
• nCk{}_nC_k — the form most calculator displays use

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.

See All Combinatorics Symbols and Notations →


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)