--- title: "Permutations and Combinations" description: "Count arrangements where order matters and selections where it does not, and see the second as the first divided by the arrangements you refuse to distinguish." canonical: https://lightmysky.com/learn/mathematics/permutations-and-combinations-mt_1L58kuu-w3 source: https://lightmysky.com/learn/mathematics/permutations-and-combinations-mt_1L58kuu-w3.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Permutations and Combinations Count arrangements where order matters and selections where it does not, and see the second as the first divided by the arrangements you refuse to distinguish. Subject: Mathematics · Area: Discrete Mathematics · Ages 18 to 19 Page: https://lightmysky.com/learn/mathematics/permutations-and-combinations-mt_1L58kuu-w3 ## Ready when they can - Choose between a permutation and a combination count from the wording of a problem - Derive the combination formula from the permutation count by dividing out repeats - Count arrangements of a multiset where some objects are identical ## Lesson: Order matters, or it does not A permutation is a list where order matters: medals, passwords, and officer posts all change when you reorder them. A combination is a group where order does not matter: committees, card hands, and toppings stay the same however you list them. Watch the wording: ranked, arranged, or seated in a row shouts permutation, while chosen or selected whispers combination. **Example.** Ten students run for president, vice president, and secretary, three different posts. Anyone of the 10 can take the first post, then 9 remain for the second, then 8 for the third. The count is 10 times 9 times 8, or 720 ordered selections. A combination starts from that permutation count and divides out the repeats. Each chosen group was counted once per rearrangement, and 3 people rearrange in 3 times 2 times 1, or 6 ways. So a 3-person committee from 10 students is 720 divided by 6, or 120. Symmetry helps: choosing 5 books to shelve from 7 equals choosing the 2 to leave out, so C(7, 5) equals C(7, 2), which is 21. **Tip.** Identical objects need the same treatment: divide by each repeat factorial. The word LEVEL has 5 letters with E twice and L twice, so its arrangements are 5 factorial divided by 2 factorial times 2 factorial, or 120 divided by 4, which is 30. **Recap.** Permutations count orders, combinations divide the order away, and repeats divide again. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Counting with the Product and Sum Rules](https://lightmysky.com/learn/mathematics/counting-with-the-product-and-sum-rules-mt_3NZC5F7RKP) ## Opens up - [Binomial Coefficients and Combinatorial Identities](https://lightmysky.com/learn/mathematics/binomial-coefficients-and-combinatorial-identities-mt_lHXFvudTh6)