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.

Voorbeeld van werk: 5 choose 2

Discrete-Math · Counting

5 choose 2

5,\ 2

Stap voor stap

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

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

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

Onthul het antwoord
\binom{5}{2} = 10

Symbols used here

\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
n!
factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\emptyset,\ |A|
empty set, cardinality
The set with no elements; the number of elements of A.

How to: 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.

Questions people ask

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.

Probeer je eigen

Meer 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