Combinatorics and Probability Formulas

Permutations, arrangements and combinations, classical probability, the probability of a complementary event and of independent events, Bernoulli's formula and the inclusion–exclusion principle for two and three sets.

9 formulas · Grades 9–11

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All 9 formulas on this page
  1. Number of permutations
  2. Number of combinations C(n,k)
  3. Number of permutations A(n,k)
  4. Classical probability of an event
  5. Probability of the complementary event
  6. Inclusion–exclusion principle (for two sets)
  7. Inclusion–exclusion principle (for three sets)
  8. Probability of independent events
  9. Bernoulli's formula

Number of permutations

Grade 9
Pn=n!P_n = n!

Number of ways to arrange n elements (n factorial).

Legend:
  • n — Number of elements n (integer ≥ 0)

Number of combinations C(n,k)

Grade 9
Cnk=n!k!(n−k)!C_n^k = \frac{n!}{k!(n-k)!}

Number of ways to choose k elements out of n, order not counted.

Legend:
  • n — Number of elements (n)
  • k — Number chosen (k)

Number of permutations A(n,k)

Grade 9
Ank=n!(n−k)!A_n^k = \frac{n!}{(n-k)!}

Number of ways to choose and order k elements out of n.

Legend:
  • n — Number of elements (n)
  • k — Number chosen (k)

Classical probability of an event

Grade 9
P(A)=mnP(A) = \frac{m}{n}

Probability of an event as the ratio of favourable outcomes to all outcomes.

Legend:
  • m — Favourable outcomes (m)
  • n — Total outcomes (n)

Probability of the complementary event

Grade 9
P(Aˉ)=1−P(A)P(\bar A) = 1 - P(A)

Probability that event A does not occur.

Legend:
  • p — Probability of event P(A)

Inclusion–exclusion principle (for two sets)

Grade 9
∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A \cup B| = |A| + |B| - |A \cap B|

Determines the number of elements in the union of two finite sets.

Legend:
  • A — Cardinality of set |A|
  • B — Cardinality of set |B|
  • AB — Cardinality |A∩B|

Inclusion–exclusion principle (for three sets)

Grade 9
∣A∪B∪C∣=∣A∣+∣B∣+∣C∣−∣A∩B∣−∣A∩C∣−∣B∩C∣+∣A∩B∩C∣|A \cup B \cup C| = |A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|

Determines the number of elements in the union of three finite sets.

Legend:
  • A — Cardinality of set |A|
  • B — Cardinality of set |B|
  • C — Cardinality of set |C|
  • AB — Cardinality |A∩B|
  • AC — Cardinality |A∩C|
  • BC — Cardinality |B∩C|
  • ABC — Cardinality |A∩B∩C|

Probability of independent events

Grade 11
P(A∩B)=P(A)⋅P(B)P(A \cap B) = P(A) \cdot P(B)

Probability that two independent events both occur.

Legend:
  • pa — Probability of event P(A)
  • pb — Probability of event P(B)

Bernoulli's formula

Grade 11
Pn(k)=Cnk⋅pk⋅(1−p)n−kP_n(k) = C_n^k \cdot p^k \cdot (1-p)^{n-k}

Probability of exactly k successes in n independent trials with success probability p.

Legend:
  • n — Number of elements (n)
  • k — Number chosen (k)
  • p — Probability of success (p)

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