--- title: "Binomial Coefficients and Combinatorial Identities" description: "Read the binomial coefficient as a count and prove identities about it by counting one set two ways rather than by algebra." canonical: https://lightmysky.com/learn/mathematics/binomial-coefficients-and-combinatorial-identities-mt_lHXFvudTh6 source: https://lightmysky.com/learn/mathematics/binomial-coefficients-and-combinatorial-identities-mt_lHXFvudTh6.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`. # Binomial Coefficients and Combinatorial Identities Read the binomial coefficient as a count and prove identities about it by counting one set two ways rather than by algebra. Subject: Mathematics · Area: Discrete Mathematics · Ages 18 to 19 Page: https://lightmysky.com/learn/mathematics/binomial-coefficients-and-combinatorial-identities-mt_lHXFvudTh6 ## Ready when they can - Prove Pascal's rule by splitting the selections according to whether one element is chosen - Give a counting argument for the sum of a row of Pascal's triangle - Prove Vandermonde's identity by counting a mixed selection two ways ## Lesson: One set, counted two ways The binomial coefficient C(n, r) counts the r-person groups you can form from n people. Compute it as n factorial divided by r factorial times n minus r factorial. For example, C(5, 2) is 5 times 4 divided by 2 times 1, or 10, and C(9, 2) is 9 times 8 divided by 2, or 36. A binomial coefficient always comes out a whole number. **Example.** Fix one person, say Ana, and split every pair from 6 people by whether it holds her. Pairs that skip Ana choose 2 from the other 5, and pairs that hold her choose 1 more from the other 5. That split is Pascal's rule: C(6, 2) equals C(5, 2) plus C(5, 1), or 10 plus 5, which is 15. Add a whole row of Pascal's triangle by giving each of n people a say: in or out. Each person has 2 choices, so the row sums to 2 to the power n. Row 4 reads 1, 4, 6, 4, 1, and sums to 16, which is 2 to the power 4. Vandermonde's identity counts one mixed team two ways. Take 2 juniors and 3 seniors and form 2-person teams. Directly, that is C(5, 2), or 10. Split by juniors on the team: 0 juniors with 2 seniors gives 1 times 3, 1 with 1 gives 2 times 3, and 2 with 0 gives 1 times 1. Adding gives 3 plus 6 plus 1, or 10 again. **Recap.** Read C(n, r) as a count, split selections on one person for Pascal, and count one team two ways for Vandermonde. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Permutations and Combinations](https://lightmysky.com/learn/mathematics/permutations-and-combinations-mt_1L58kuu-w3) - [The Binomial Expansion](https://lightmysky.com/learn/mathematics/the-binomial-expansion-mt_z16HZkNZ8c) ## Opens up - [The Pigeonhole Principle](https://lightmysky.com/learn/mathematics/the-pigeonhole-principle-mt_wgfpmZqzd9)