Calculate various possibilities from basic permutations and combinations to repetitions and derangements. Essential tool for probability and statistics problem solving.
nPr = n! / (n-r)!
nCr = n! / (r! × (n-r)!)
n^r (repetition allowed)
H(n,r) = C(n+r-1, r)
!n = n! × Σ((-1)^k / k!)
Permutation considers order when selecting items (AB ≠ BA), while combination does not (AB = BA). For selecting 2 from ABC: permutation gives 6 results (AB, BA, AC, CA, BC, CB), combination gives 3 (AB, AC, BC). Use permutations for passwords or race rankings where order matters, combinations for team selection or lottery numbers where order does not matter.
Permutations with repetition allow selecting the same item multiple times with order (n^r). Rolling a die 3 times has 6^3 = 216 outcomes. Combinations with repetition allow duplicates without order: H(n,r) = C(n+r-1, r). Choosing 3 ice cream scoops from 5 flavors (duplicates allowed) is a classic example. These concepts model real-world selection scenarios mathematically.
Derangement counts arrangements where no element appears in its original position. Denoted !n, calculated as !n = n! × (1/0! - 1/1! + 1/2! - ... + (-1)^n/n!). For 3 people randomly picking hats, !3 = 2 (arrangements BCA, CAB where none gets their own hat). Used in probability theory, Secret Santa games, and card shuffling problems.
Solving probability problems requires calculating total outcomes and specific event outcomes. Drawing 5 cards from 52 uses combinations for total possibilities (C(52,5)). Applied to poker flush probability, lottery odds, password cracking difficulty, and more. Correctly distinguishing permutations from combinations is the first step to accurate probability calculations.
Factorial (n!) is the product of all integers from 1 to n, fundamental to permutations and combinations. 5! = 120, 0! = 1 by definition. Factorials grow rapidly (10! = 3,628,800, 20! ≈ 2.4×10^18), requiring overflow precautions. Stirling's approximation estimates large n: n! ≈ √(2πn) × (n/e)^n. Widely used in algorithm complexity analysis and statistical distribution calculations.
Permutations and combinations appear throughout daily life: selecting 3 from 5 meeting attendees (C(5,3)=10), creating 4-digit passwords (10^4=10,000), seating 10 people in 3 chairs (P(10,3)=720), choosing 2 ice cream scoops from 5 flavors (H(5,2)=15). Essential for algorithm design, data structure analysis, probability simulation in programming. This calculator helps analyze complex real-world problems mathematically for optimal decision-making.