--- title: Combinations in Combinatorics - Unordered Selections description: Combinations count unordered selections of objects. This chapter covers n choose k and how combinations differ from permutations. url: https://astra-ai.co/videos/combinatorics/combinations/ locale: en type: video-chapter keywords: - combinations - unordered selections - binomial coefficient - n choose k published: 2026-10-01T09:48:50.932Z updated: 2026-10-04T21:58:54.513Z alternates: nl: https://astra-ai.co/nl/videos/combinatoriek/combinaties/ lo: https://astra-ai.co/lo/videos/lo-dd33883f/lo-53cb754d/ it: https://astra-ai.co/it/videos/combinatoria/combinazioni/ lt: https://astra-ai.co/lt/videos/kombinatorika/deriniai/ fi: https://astra-ai.co/fi/videos/kombinatorikka/yhdistelmat/ vi: https://astra-ai.co/vi/videos/vi-topic-05/vi-chapter-25/ fr: https://astra-ai.co/fr/videos/combinatoire/combinaisons/ pl: https://astra-ai.co/pl/videos/kombinatoryka/kombinacje/ sl: https://astra-ai.co/sl/videos/kombinatorika/kombinacije/ hr: https://astra-ai.co/hr/videos/kombinatorika/kombinacije/ pt: https://astra-ai.co/pt/videos/combinatoria/combinacoes/ zh-HK: https://astra-ai.co/zh-HK/videos/shuxue-kecheng-005/shuxue-zhang-025/ da: https://astra-ai.co/da/videos/kombinatorik/kombinationer/ el: https://astra-ai.co/el/videos/syndyastiki/syndyasmoi/ bg: https://astra-ai.co/bg/videos/kombinatorika/kombinatsii/ cs: https://astra-ai.co/cs/videos/kombinatorika/kombinace/ ko: https://astra-ai.co/ko/videos/combinatorics-ko/51/ uk: https://astra-ai.co/uk/videos/uk-topic-05/uk-chapter-25/ id: https://astra-ai.co/id/videos/kombinatorika/kombinasi/ hu: https://astra-ai.co/hu/videos/kombinatorika/kombinaciok/ et: https://astra-ai.co/et/videos/kombinatoorika/kombinatsioonid/ sv: https://astra-ai.co/sv/videos/kombinatorik/kombinationer/ ro: https://astra-ai.co/ro/videos/combinatorica/combinari/ sk: https://astra-ai.co/sk/videos/kombinatorika/kapitola-051/ no: https://astra-ai.co/no/videos/kombinatorikk/kombinasjoner/ sr: https://astra-ai.co/sr/videos/kombinatorika/poglavlje-051/ ms: https://astra-ai.co/ms/videos/combinatorics/bab-25-matematik-asas/ lv: https://astra-ai.co/lv/videos/kombinatorika/kombinacijas/ tr: https://astra-ai.co/tr/videos/kombinatorik/bolum-051/ zh-TW: https://astra-ai.co/zh-TW/videos/combinatorics/chapter-53cb754d/ th: https://astra-ai.co/th/videos/combinatorics/bot-25-kan-chat-mu/ my: https://astra-ai.co/my/videos/my-dd33883f/my-53cb754d/ km: https://astra-ai.co/km/videos/combinatorics/chapter-53cb754d/ en: https://astra-ai.co/videos/combinatorics/combinations/ es-AR: https://astra-ai.co/es-AR/videos/combinatoria/capitulo-51/ br: https://astra-ai.co/br/videos/combinatoria/capitulo-matematica-025/ pt-TL: https://astra-ai.co/pt-TL/videos/combinatoria/capitulo-de-video-051/ mx: https://astra-ai.co/mx/videos/combinatoria/capitulo-51/ en-IN: https://astra-ai.co/en-IN/videos/combinatorics/chapter-53cb754d/ es-CL: https://astra-ai.co/es-CL/videos/combinatoria/capitulo-51/ ms-BN: https://astra-ai.co/ms-BN/videos/kombinatorik/bab-53cb754d/ es-PE: https://astra-ai.co/es-PE/videos/combinatoria/capitulo-video-53cb754d/ en-PH: https://astra-ai.co/en-PH/videos/combinatorics/chapter-53cb754d/ ja: https://astra-ai.co/ja/videos/combinatorics-ja/51/ de: https://astra-ai.co/de/videos/kombinatorik/kombinationen/ es: https://astra-ai.co/es/videos/combinatoria/combinaciones/ --- # Combinations > Combinations count unordered selections of objects. This chapter covers n choose k and how combinations differ from permutations. ## INTRODUCTION TO UNORDERED SELECTIONS In combinatorics, **combinations** represent a way of selecting elements from a given set where the **order does not matter**. This means that selections like {A, B, C} and {C, B, A} are considered the same combination. Combinations are often used to count possibilities in cases where order is irrelevant – for example, when drawing lottery numbers, selecting people for a group, or choosing dishes from a menu. If we select k elements from a set of n distinct elements, where the order does not matter and elements are chosen without repetition, we are talking about **combinations without repetition**. ## COMBINATIONS WITHOUT REPETITION The number of all such combinations is denoted by C(n,k) or (kn​), which is calculated by the formula: C(n,k)=n!/\[k!∗(n−k)!] This formula accounts for all possible arrangements (n!/(n−k)!) and then divides them by the number of possible orderings of the k chosen elements (k!), since we are **not interested** in the order. ### Example: Calculating Combinations Without Repetition How many different 3-person teams can be formed from 5 people? n=5 (total people), k=3 (team members) C(5,3)=5!/(3!∗2!)=120/(6∗2)=10 There are 10 different 3-person teams that can be formed from 5 people. ## COMBINATIONS WITH REPETITION If we **allow repetition** of elements in the selection, we use **combinations with repetition**, denoted as C′(n,k). The formula is: C′(n,k)=C(n+k−1,k)=(n+k−1)!/\[k!∗(n−1)!] This means we have more possibilities since the same element can be chosen multiple times. ### Example: Calculating Combinations With Repetition How many ways are there to choose 4 fruits (k=4) from 3 different types of fruit (n=3), if repetition is allowed? C′(3,4)=C(3+4−1,4)=C(6,4)=15 There are 15 ways to choose 4 fruits from 3 different types when repetition is allowed. ## COMBINATIONS VS. VARIATIONS: KEY DIFFERENCES Understanding the distinction between combinations and variations is fundamental in combinatorics: - **Combinations:** The order of selected elements **does not** matter. - **Variations:** The order of selected elements **does** matter. Both types also have forms **with and without repetition**. ### Illustrative Example from the Set {A, B, C}: Let's consider choosing k=2 elements from the set {A, B, C}: - **Combinations (k=2):** AB, AC, BC → 3 possibilities (Note: BA is the same as AB) - **Variations (k=2):** AB, BA, AC, CA, BC, CB → 6 possibilities (Note: BA is different from AB) ## PRACTICAL USES OF COMBINATIONS Combinations are applied in various real-world scenarios, including: - **Lottery draws:** Calculating the probability of winning based on selecting numbers where order doesn't matter. - **Forming committees, teams, or groups:** Where the arrangement of individuals within the group is not relevant. - **Choosing items where sequence is unimportant:** Such as picking toppings for a pizza or selecting books from a shelf. - **Probability calculations:** Especially in scenarios involving sampling without replacement where the sequence of selection doesn't affect the outcome. ## CONCLUSION: MASTERING COMBINATORIAL COUNTING Combinations are a foundational counting method for cases where we select **without considering order**. The number of all combinations is always less than the number of variations from the same set, as permutations within the same group are counted as a single instance. Knowing the differences between combinations, permutations, and variations is crucial for accurate counting in various combinatorial situations.