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The counting principles
Product rule, sum rule, permutations, combinations, and when each applies.
Independent choices multiply; exclusive alternatives add. Ordered selections are permutations, unordered are combinations. Picture it: a tree diagram, each level a choice, each leaf an outcome. Think it: every counting formula is a bijection with something easy to count.
Ví dụ đã làm · 5 choose 2
Bước một bước
- \binom{5}{2} = \frac{5!}{2!\,(5-2)!}
Unordered selections: n! / (k! (n−k)!).
- = \frac{120}{2 \times 6} = 10
Giải đáp
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Làm thế nào: The counting principles
- Decide whether order matters (permutation) or not (combination).
- Decide whether repetition is allowed.
- Break the count into independent stages (multiply) or exclusive cases (add).
- Sanity-check with a tiny case you can list by hand.
Câu hỏi người ta hỏi
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.
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Pigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesPaths, cycles, trees, Euler and HamiltonColouring and planar graphs