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Permutations and Combinatorics

Permutation formulae

  • Permutation of n objects with duplicates
\[ N = \frac{n!}{p!q!r!\ldots} \]
  • Circular Permutation

Strategy: Fix one of the objects and permute the rest. Alternatively, permute the ring as normal but divide by N at the end, due to the N-order symmetry.

\[ N = (n-1)! \]

Probability relations

  • Independence: \(P(A\cap B) = P(A)P(B)\)
  • Mutually exclusive: \(P(A \cap B) = 0\)
  • Conditional probability (Probability of A given B):
\[ P(A | B) = \frac{P(A\cap B)}{P(B)} \]