maths.free › Combinatorics & Graph Theory › The counting principles

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.

Vinna dæmi · 5 choose 2

Discrete-Math · Counting

5 choose 2

5,\ 2

Skref fyrir skref

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

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

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

Sýna svarið
\binom{5}{2} = 10

Núna ūú Veldu vandamál, eða sláðu inn eða teikna eigin. Hvert skref, mynd, svarið falið þar til þú spyrð.

Haltu eigin vinnu þinni

Ókeypis reikningur bætir við athugasemdum við hverja kennslustund, skrá yfir það sem þú hefur lokið, leyst vandamál á einum stað og kennari sem þú getur spurt um þessa síðu.Stærðfræðin sjálf er opin öllum, skráður inn eða ekki.

Skráðu þig inn Innskráning

Tákn notuð hér

Pikkaðu á hvaða tákn sem er fyrir fulla skilgreiningu, mynd og hvað hvert bréf í því merkir.

Hvernig á að nota: 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.

Spurningar sem fólk spyr

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.

Meira í 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