Calculate permutations (nPr) and combinations (nCr) instantly with formulas.
A permutation and combination calculator instantly computes nPr (permutations) and nCr (combinations) for any values of n and r. These calculations are fundamental in probability theory, statistics, and competitive mathematics — and appear regularly in exams like JEE, CAT, GRE, and GMAT. Our nCr calculator handles large values that are impossible to calculate by hand.
Permutation (nPr) counts the number of ways to arrange r items from n items where the order of arrangement matters. Formula: nPr = n! / (n-r)!. Example: How many 3-digit numbers can be formed from {1,2,3,4,5} without repetition? = 5P3 = 5!/(5-3)! = 120/2 = 60. Each different arrangement (123, 132, 213...) is a separate permutation.
Combination (nCr) counts the number of ways to select r items from n items where order does not matter. Formula: nCr = n! / (r! × (n-r)!). Example: How many ways to choose 3 team members from 8 candidates? = 8C3 = 8!/(3!×5!) = 56. Choosing Alice, Bob, Charlie is the same combination regardless of order.
Lottery odds: Winning a 6/49 lottery = choosing 6 numbers from 49 = 49C6 = 13,983,816 combinations. Password security: An 8-character password with 26 lowercase letters has 26^8 = 208 billion permutations. Sports scheduling: Round-robin tournament matches between n teams = nC2 matches. Committee selection: Choosing 5 from 20 people = 20C5 = 15,504 ways.
Pascal's Triangle is a triangular array where each number is the sum of the two numbers above it. The nth row contains the combinations nC0, nC1, nC2, ... nCn. Row 4: 1, 4, 6, 4, 1 (which are 4C0, 4C1, 4C2, 4C3, 4C4). Pascal's Triangle connects combinations to the binomial theorem: (a+b)^n = Σ nCr × a^(n-r) × b^r.
Permutation considers the order of selection (AB ≠ BA), while combination does not (AB = BA). Use permutation when order matters.
Use permutation when arranging items where order matters, like ranking winners in a race or creating passwords.
Use combination when selecting items where order does not matter, like choosing team members or selecting lottery numbers.
nPr = n! / (n-r)! where n is total items and r is items being selected. The ! symbol means factorial.
