From an m-element set A to an n-element set B: Total functions = $n^m$ (each of m elements has n choices). Injective functions = P(n,m) = n!/(n-m)! (requires m <= n; first element has n choices, second n-1, etc.). Surjective functions (m >= n): use inclusion-exclusion = $sum_{k=0}^{n}$ (-1)^k * C(n,k) * (n-k)^m. Bijective functions = n! (requires m = n). Example: from {1,2,3,4} to {a,b}: total = 2^4 = 16, surjective = 16 - 2 = 14 (subtract functions mapping everything to a single element). Number of bijections from {1,2,3} to {a,b,c} = 3! = 6. These counting formulas are tested both directly and within probability problems.