Ten days of statistics (4) - Combinatorics

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Permutations

A permutation is an ordered arrangement of rr objects from a set of nn items (0<r≤n0< r \leq n)

nPr=n!(n−r)!_nP_r = \frac{n!}{(n-r)!}

Note: We define 0!=10! = 1

Combinations

A combination is an unordered arrangement of rr objects from a set of nn items (0<r≤n0< r \leq n) Intuitively we can calculate combination from permutation

nCr=nPrr!=n!r!×(n−r)! _nC_r = \frac{_nP_r}{r!} = \frac{n!}{r! \times (n-r)!}

Example of permutation and combination

  • A school need to pick 11 players to form a team out of 100 candidates. This is a combination, there are C(10011)C \binom{100}{11} ways to form the team
  • A contest has 100 contestants and 3 medals (gold, silver, bronze) for 3 best performers. This is a permutation, there are P(1003)P \binom{100}{3} ways for the contest to end

Next lesson: Binomial distribution