Variations Calculator

Definition

Let n and k be non-negative integers. A variation without repetition V(n,k) — in English combinatorics usually P(n,k), a “k-permutation of n”, or the number of injections from a k-set into an n-set — is the number of ordered k-tuples of pairwise distinct entries chosen from an n-element set. The classical formula is V(n,k) = n! / (n−k)! for 0 ≤ k ≤ n. Equivalently the falling factorial n^{\underline{k}} = n·(n−1)·…·(n−k+1). When k > n no such tuple exists without repetition, so this calculator returns 0.

Polish school texts prefer “wariacja”; English sources avoid “variation” as a calque and write P(n,k). MathWorld (Permutation) and textbooks by Brualdi and Concrete Mathematics (Graham, Knuth, Patashnik) treat P(n,k) as the injection count or as the number of ways to fill k ordered seats from n candidates without reuse.

This page does not divide full factorials for large n directly: it computes ∏_{i=0}^{k−1} (n − i) and also shows n! and (n−k)! in the panel as a check of the quotient form. For typical school data the result matches the definition.

Historical and conceptual context

The multiplication rule (“n choices for the first seat, n−1 for the second…”) appears in medieval and early modern texts long before modern factorial notation. Christian Kramp (1760–1826) popularized the symbol n! for 1·2·…·n. Nineteenth-century enumerative combinatorics separated full orders (permutations), partial orders (k-permutations / variations), and unordered selections (combinations).

Conceptually, variations sit between permutations and combinations. Permutations P(n) = n! arrange all items; combinations C(n,k) = n!/(k!(n−k)!) pick a set without order; variations V(n,k) = n!/(n−k)! arrange k of n with order. Bridge: V(n,k) = C(n,k)·k!. In computer science P(n,k) counts injections, codes without repeated symbols, and drives partial-permutation generators. In probability it appears in ordered sampling without replacement.

This calculator does not solve for unknown n or k, does not list tuples, and does not count variations with repetition (that formula is n^k). It answers one question: how many ordered picks of k from n without repetition.

Formal properties

Domain on this page: n, k ∈ ℕ₀. For 0 ≤ k ≤ n: V(n,k) = n!/(n−k)! = n^{\underline{k}}. V(n,0) = 1. V(n,1) = n. V(n,n) = n!. V(n,k) = 0 when k > n. Recurrence: V(n,k) = n · V(n−1,k−1) for k ≥ 1 (with V(m,0) = 1). Alternatively V(n,k) = (n−k+1) · V(n,k−1) for k ≥ 1.

Link to combinations: V(n,k) = C(n,k)·k! and C(n,k) = V(n,k)/k!. Link to full permutations: V(n,n) = P(n) = n!. For fixed n ≥ 1 the sequence in k usually rises to V(n,n) = n!.

The quotient identity V(n,k)/V(n,k−1) = n−k+1 for 1 ≤ k ≤ n ties neighboring lengths. Variations with repetition n^k do not divide by the factorial of unused items — a different model.

Full derivation

Step 1 (multiplication rule). The first of k ordered seats has n choices, the second has n−1 remaining, …, the k-th has n−k+1. The product n·(n−1)·…·(n−k+1) is V(n,k).

Step 2 (factorials). Full factorial n! counts all orders of n items. Orders of the unused n−k items are (n−k)!. Dividing n! by (n−k)! strips the orders you never arrange: V(n,k) = n!/(n−k)!.

Step 3 (bridge to C). Each unordered k-set corresponds to exactly k! orderings. Hence C(n,k) = V(n,k)/k! and V(n,k) = C(n,k)·k!.

Step 4 (boundaries). For k = 0 the empty product is 1. For k = n the denominator is 0! = 1 and V(n,n) = n!. For k > n the falling product has no meaning without repetition — result 0.

Step 5 (examples). V(5,3) = 5·4·3 = 60 = 120/2. V(6,2) = 6·5 = 30. V(4,4) = 24. V(3,5) = 0.

Special cases

V(n,2) = n(n−1) — ordered pairs of distinct items. A top-3 ranking from 20 athletes: V(20,3) = 20·19·18 = 6,840. A 4-letter code from 26 distinct letters: V(26,4) = 26·25·24·23 = 358,800 (not the with-repetition model).

Negative n or k after flooring are rejected with an error (undefined). Non-integers are floored: 5.8 and 3.2 give V(5,3) = 60. When k > n → 0 (not a form error).

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 falling product limits magnitude growth versus a direct n! / (n−k)! split for large n and small k. The panel still shows n! and (n−k)! as an educational breakdown — for large n those factorials may become Infinity while V stays finite. Comma and period are equivalent.

The bar chart is limited to n ≤ 10, and the seat diagram to moderate n and k, so graphics stay readable. C(n,k) in the panel is the check V(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), combinations C(n,k), variations with repetition n^k, falling factorial, finite injections, codes without repeated symbols. The identity V(n,k) = C(n,k)·k! ties the three classic school counts.

Sources: Brualdi, Introductory Combinatorics (permutations / P(n,k)); Graham, Knuth, Patashnik, Concrete Mathematics (falling factorials); MathWorld — Permutation; Wikipedia — Permutation / k-permutation. Educational tool — not a substitute for a textbook or teacher check.