Combinations Calculator

Definition

Let n and k be non-negative integers. The binomial coefficient C(n,k), also written \(\binom{n}{k}\) or “n choose k”, is the number of k-element subsets of an n-element set. Equivalently: the number of ways to choose k distinct items from n when order does not create a new outcome. The classical formula is C(n,k) = n! / (k! · (n−k)!) for 0 ≤ k ≤ n; outside that range the model without repetition has no legal selection (this calculator reports an error when k > n instead of returning 0).

The name “binomial coefficient” comes from the binomial theorem: (a + b)^n = Σ_{k=0}^{n} C(n,k) a^{n−k} b^k. NIST DLMF §26.3 collects identities; MathWorld (Binomial Coefficient) and Wikipedia use the same set-counting definition. Polish school texts often say “Newton’s symbol” for the same C(n,k).

This page does not multiply full factorials for large n directly: it computes ∏_{i=1}^{r} (n − r + i) / i with r = min(k, n − k), then round to the nearest integer. For typical school data the result matches the definition.

Historical and conceptual context

Counting unordered selections appears in Indian and Arabic mathematics earlier than the modern factorial form. Blaise Pascal (1623–1662) systematized the arithmetic triangle now called Pascal’s triangle: each entry is the sum of two entries above. Isaac Newton linked the coefficients to the binomial expansion, including non-integer exponents (binomial series). The notations C(n,k) and \(\binom{n}{k}\) spread with nineteenth- and twentieth-century combinatorics.

Conceptually, combinations sit beside permutations and variations. Permutations P(n) = n! arrange all items; variations without repetition V(n,k) = P(n,k) = n!/(n−k)! arrange k of n with order; combinations C(n,k) pick a set without order. Bridge identity: P(n,k) = C(n,k) · k!. In probability, C(n,k) counts ways to get exactly k successes in n Bernoulli trials. In computer science it counts k-subsets and drives combination-generation complexity.

This calculator does not solve for unknown n or k, does not list subsets, and does not count combinations with repetition (that formula is C(n + k − 1, k)). It answers one question: how many unordered picks of k from n without repetition.

Formal properties

Domain on this page: n, k ∈ ℕ₀, 0 ≤ k ≤ n. C(n,0) = C(n,n) = 1. Symmetry: C(n,k) = C(n,n−k). Pascal’s identity: C(n,k) = C(n−1,k−1) + C(n−1,k) for 1 ≤ k ≤ n − 1. Row sum: Σ_{k=0}^{n} C(n,k) = 2^n. Product identity: C(n,k) · C(k,m) = C(n,m) · C(n−m,k−m) when 0 ≤ m ≤ k ≤ n.

Link to factorial and variations: C(n,k) = n! / (k!(n−k)!) = P(n,k) / k!. Multiplicative recurrence: C(n,k) = C(n,k−1) · (n − k + 1) / k for k ≥ 1. For fixed n, values rise toward the middle of the row and fall symmetrically.

In the binomial theorem the coefficients are non-negative integers; Σ C(n,k) = 2^n counts all subsets of an n-set.

Full derivation

Step 1. The number of ordered picks of k distinct items from n is P(n,k) = n · (n−1) · … · (n−k+1) = n! / (n−k)!. Each unordered set {x₁,…,x_k} corresponds to exactly k! orderings. Hence C(n,k) = P(n,k) / k! = n! / (k!(n−k)!).

Step 2 (symmetry). A bijection between k-subsets and (n−k)-subsets of complements gives C(n,k) = C(n,n−k). So the calculator may use r = min(k, n−k).

Step 3 (Pascal). The k-subsets of {1,…,n} split into those that contain n (completed in C(n−1,k−1) ways) and those that do not (C(n−1,k) ways). The sum is C(n,k).

Step 4 (product). C(n,k) = ∏_{i=1}^{k} (n − k + i) / i with on-the-fly cancellation avoids huge intermediate factorials. Each step is reduced by the GCD, so results up to 2⁵³ are exact; larger values are shown as an approximation ≈ m × 10^e.

Step 5 (examples). C(5,2) = (5·4)/(1·2) = 10. C(10,3) = (10·9·8)/(1·2·3) = 120. C(8,0) = 1. C(6,6) = 1.

Special cases

C(n,1) = n; C(n,2) = n(n−1)/2 (edges of the complete graph K_n). C(52,5) = 2,598,960 — five-card hands from a 52-card deck. A 6-from-49 lottery: C(49,6) = 13,983,816. For n = 0: C(0,0) = 1.

Negative n or k after flooring are rejected with an error. Non-integers are floored: 5.7 and 2.1 give C(5,2) = 10. When k > n → error (not zero).

For very large n, floating-point arithmetic loses integer exactness; larger n needs big-integer arithmetic. This educational calculator targets the school range and the panel examples.

Numerical remarks

The multiplicative product with division at each step limits magnitude growth versus n! / (k!(n−k)!). Final round matches the earlier gold copy (“rounds to an integer”). Comma and period are equivalent.

The Pascal chart is limited to n ≤ 12 so bars stay readable. P(n,k) in the panel is the check C(n,k)·k!, not a separate form mode.

Numbers are not sent to a server; the browser computes everything in floating-point arithmetic.

Related notions and sources

Related: factorial n!, permutations P(n), variations V(n,k) = P(n,k), Pascal’s triangle, binomial theorem, binomial distribution, combinations with repetition. Polygon diagonals n(n−3)/2 = C(n,2) − n.

Sources: NIST DLMF §26.3 (Binomial Coefficients); MathWorld — Binomial Coefficient; Wikipedia — Binomial coefficient; Pascal, Traité du triangle arithmétique; Feller, An Introduction to Probability Theory. Educational tool — not a substitute for a textbook or teacher check.