--- title: What Are Variations in Combinatorics? (k-Permutations) description: Variations in combinatorics count ordered selections, with or without repetition. See how they differ from permutations and combinations. url: https://astra-ai.co/videos/combinatorics/variations-in-combinatorics/ locale: en type: video-chapter keywords: - variations in combinatorics - ordered selections - repetition - counting arrangements published: 2026-10-01T09:48:31.730Z updated: 2026-10-04T21:58:48.193Z alternates: my: https://astra-ai.co/my/videos/my-dd33883f/my-4e1a600e/ fi: https://astra-ai.co/fi/videos/kombinatorikka/variaatiot-kombinatoriikassa/ de: https://astra-ai.co/de/videos/kombinatorik/variationen/ es: https://astra-ai.co/es/videos/combinatoria/variaciones-en-combinatoria/ fr: https://astra-ai.co/fr/videos/combinatoire/variations-en-combinatoire/ sl: https://astra-ai.co/sl/videos/kombinatorika/variacije/ hr: https://astra-ai.co/hr/videos/kombinatorika/varijacije/ pt: https://astra-ai.co/pt/videos/combinatoria/variacoes/ nl: https://astra-ai.co/nl/videos/combinatoriek/variaties/ da: https://astra-ai.co/da/videos/kombinatorik/variationer-i-kombinatorik/ el: https://astra-ai.co/el/videos/syndyastiki/diatakseis-sti-syndyastiki/ bg: https://astra-ai.co/bg/videos/kombinatorika/variatsii/ cs: https://astra-ai.co/cs/videos/kombinatorika/variace/ hu: https://astra-ai.co/hu/videos/kombinatorika/variaciok-a-kombinatorikaban/ id: https://astra-ai.co/id/videos/kombinatorika/variasi-dalam-kombinatorika/ et: https://astra-ai.co/et/videos/kombinatoorika/variatsioonid-kombinatoorikas/ ja: https://astra-ai.co/ja/videos/combinatorics-ja/47/ sv: https://astra-ai.co/sv/videos/kombinatorik/variationer-inom-kombinatorik/ no: https://astra-ai.co/no/videos/kombinatorikk/variasjoner-i-kombinatorikk/ sk: https://astra-ai.co/sk/videos/kombinatorika/kapitola-047/ ro: https://astra-ai.co/ro/videos/combinatorica/variatii-in-combinatorica/ vi: https://astra-ai.co/vi/videos/vi-topic-05/vi-chapter-17/ uk: https://astra-ai.co/uk/videos/uk-topic-05/uk-chapter-17/ sr: https://astra-ai.co/sr/videos/kombinatorika/poglavlje-047/ ms: https://astra-ai.co/ms/videos/combinatorics/bab-17-matematik-asas/ lv: https://astra-ai.co/lv/videos/kombinatorika/variasijas-kombinatorika/ lt: https://astra-ai.co/lt/videos/kombinatorika/variacijos-kombinatorikoje/ tr: https://astra-ai.co/tr/videos/kombinatorik/bolum-047/ zh-TW: https://astra-ai.co/zh-TW/videos/combinatorics/chapter-4e1a600e/ th: https://astra-ai.co/th/videos/combinatorics/bot-17-kan-chat-mu/ km: https://astra-ai.co/km/videos/combinatorics/chapter-4e1a600e/ lo: https://astra-ai.co/lo/videos/lo-dd33883f/lo-4e1a600e/ es-AR: https://astra-ai.co/es-AR/videos/combinatoria/capitulo-47/ zh-HK: https://astra-ai.co/zh-HK/videos/shuxue-kecheng-005/shuxue-zhang-017/ br: https://astra-ai.co/br/videos/combinatoria/capitulo-matematica-017/ pt-TL: https://astra-ai.co/pt-TL/videos/combinatoria/capitulo-de-video-047/ en-IN: https://astra-ai.co/en-IN/videos/combinatorics/chapter-4e1a600e/ es-CL: https://astra-ai.co/es-CL/videos/combinatoria/capitulo-47/ mx: https://astra-ai.co/mx/videos/combinatoria/capitulo-47/ ms-BN: https://astra-ai.co/ms-BN/videos/kombinatorik/bab-4e1a600e/ es-PE: https://astra-ai.co/es-PE/videos/combinatoria/capitulo-video-4e1a600e/ en-PH: https://astra-ai.co/en-PH/videos/combinatorics/chapter-4e1a600e/ ko: https://astra-ai.co/ko/videos/combinatorics-ko/47/ en: https://astra-ai.co/videos/combinatorics/variations-in-combinatorics/ it: https://astra-ai.co/it/videos/combinatoria/variazioni-combinatoria/ pl: https://astra-ai.co/pl/videos/kombinatoryka/wariacje-w-kombinatoryce/ --- # Variations in Combinatorics > Variations in combinatorics count ordered selections, with or without repetition. This chapter shows how they differ from permutations and combinations. ## INTRODUCTION TO ORDERED SELECTIONS In combinatorics, **variations** are used to count the possible ways to **arrange a chosen number of elements** from a larger set **where the order is important**. This approach is crucial for tasks like ranking, coding, creating passwords, or seating people, especially when it matters who is first, second, third, and so on. With variations, it's not necessary to use all elements of the set. From a set with **n** elements, we choose **k** elements (where k ≤ n) and arrange them in an ordered sequence. ## VARIATIONS WITHOUT REPETITION When we do **not allow the repetition** of elements, we are talking about **variations without repetition**. The number of these variations is denoted by V(n, k) and is calculated using the formula: V(n, k) = n \* (n – 1) \* (n – 2) \* … \* (n – k + 1) Or, written with factorials: V(n, k) = n! / (n – k)! **EXAMPLE**: How many different two-digit numbers can be made using the digits 1 through 5, without repetition? - n = 5 (number of available digits) - k = 2 (we are forming two-digit numbers) V(5, 2) = 5 \* 4 = **20** ## VARIATIONS WITH REPETITION If **repetition is allowed**, then each of the k positions can be filled with any of the n elements, regardless of what has already been chosen. The formula for variations with repetition is: V'(n, k) = n^k **EXAMPLE**: How many different two-digit numbers can be made using the digits 1 through 5 if repetition is allowed? - n = 5 - k = 2 V'(5, 2) = 5^2 = **25** ## DIFFERENCE BETWEEN VARIATIONS AND PERMUTATIONS - In **permutations**, we arrange **all** the elements (k = n). - In **variations**, we arrange **only a subset** of the elements (k < n). - In both cases, the **order is important**. **EXAMPLE**: Given the set {A, B, C}, so n = 3: - **Permutations** (k = 3): ABC, ACB, BAC, BCA, CAB, CBA → 3! = **6** - **Variations** without repetition (k = 2): AB, AC, BA, BC, CA, CB → V(3, 2) = **6** ## PRACTICAL USES Variations appear in various contexts: - Determining possible passwords or codes. - Ranking finalists based on their finishing positions. - Forming combinations of cards, numbers, or symbols where order plays a role. ## CONCLUSION Variations are a fundamental tool in combinatorics for handling the **ordered selection** of a smaller number of elements from a given set. We distinguish between cases **with and without repetition**, but in every scenario, the key takeaway is that the order of the selected elements is not negligible and directly impacts the final result.