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Binomial coefficients and Pascal's triangle
Identities, the binomial theorem, and combinatorial proofs.
C(n, k) counts k-subsets, is the coefficient in (a + b)ⁿ, and satisfies Pascal's rule C(n, k) = C(n−1, k−1) + C(n−1, k). Picture it: Pascal's triangle, each entry the sum of the two above. Think it: a combinatorial proof shows two formulas count the same set, no algebra needed.
Arbejdstxrd eksempel · expand (a + b)^6
Trin for trin
- \left(a + b\right)^{6}
Start from the expression.
- a^{6} + 6 a^{5} b + 15 a^{4} b^{2} + 20 a^{3} b^{3} + 15 a^{2} b^{4} + 6 a b^{5} + b^{6}
Distribute (multiply out) the products.
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Hvordan: Binomial coefficients and Pascal's triangle
- Start from the expression.
- Distribute (multiply out) the products.
Spørgsmål folk spørger
Permutation or combination?
Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).
What is a graph in this sense?
Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.
Mere i Combinatorics & Graph Theory
The counting principlesPigeonhole principle and inclusion, exclusionRecurrences and generating functionsGraphs: vertices, edges, degreesPaths, cycles, trees, Euler and HamiltonColouring and planar graphs