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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.

Urugero · 5 choose 2

Discrete-Math · Counting

5 choose 2

5,\ 2

ku

  1. \binom{5}{2} = \frac{5!}{2!\,(5-2)!}

    Unordered selections: n! / (k! (n−k)!).

  2. = \frac{120}{2 \times 6} = 10

i
\binom{5}{2} = 10

Guhitamo A, Cyangwa Ubwoko: Cyangwa Gushushanya. Intera, A Ishusho, i Gihishwe & Kugeza & Kubaza.

Kuri Gushyiraho

A Kigenga Konti: & OYA & Ongera Ibisobanuro: ku, A Ibijyana Bya, in Rimwe, na A Kubaza Bigyanye iyi Ipaji:. Imibare ni Gufungura Kuri, in Cyangwa OYA.

Kwiyandikisha Ifashayinjira

Byakoreshejwe

Icyo ari cyo cyose Ibimenyetso ya: i Byuzuye Insobanuro:, A Ishusho, na buri Ibaruwa: in.

Kuri: The counting principles

  1. Decide whether order matters (permutation) or not (combination).
  2. Decide whether repetition is allowed.
  3. Break the count into independent stages (multiply) or exclusive cases (add).
  4. Sanity-check with a tiny case you can list by hand.

Abaturage

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.

in Combinatorics & Graph Theory

Pigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesPaths, cycles, trees, Euler and HamiltonColouring and planar graphs