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Factorials apply only to nonnegative integers but can be extended to complex numbers with positive real parts using the Gamma function, \(\Gamma(n) = \int_0^\infty e^{-x}x^{n-1}dx\).

The binomial theorem, making use of the well-known binomial coefficients, allows expressing \((x+y)^n\) as a sum: \((x+y)^n=\sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}\).

The total probability formula, \(P(A) = \sum_k P(A \cap B_k)\), relates the occurrence of an event \(A\) to that of several mutually exclusive events (\(B_k\)'s).