Permutations and Combinations

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Combinatorics: Permutations vs Combinations
Permutation: ordered arrangement of objects; e.g., P(5,2)=5*4=20.
Formula: P(n, k) = n! / (n - k)!
Combination: selection without order; e.g., C(5,2)=10.
Formula: C(n, k) = n! / (k! (n - k)!)
Example: number of 5-card poker hands: C(52,5) ≈ 2.6 million.
Multiset permutations: arrangements with repeated elements: n! / (n1! n2!...).
Binomial theorem: (x+y)^n = Σ C(n, k) x^(n-k) y^k.
Applications: probability, statistics, cryptography, and algorithm analysis.