🔢 Permutation & Combination Calculator

What is a Permutation and Combination Calculator?

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.

For example, working out 3C5 or checking nPr for competitive exam problems both follow the same underlying formulas, just applied to different values of n and r.

Permutation — When Order Matters

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 — When Order Doesn't Matter

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.

Real-World Applications

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 and Combinations

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.

Permutations vs. Combinations — The Key Difference

Permutations count the number of ways to arrange items where order matters, while combinations count the number of ways to select items where order doesn't matter — this single distinction is the source of most confusion when first learning these concepts, since the same set of items can have dramatically different permutation and combination counts depending on whether arrangement order is relevant to the specific problem being solved.

A useful test for distinguishing the two: if rearranging the same selected items would create a meaningfully different outcome (like assigning first, second, and third place in a race), it's a permutation problem; if the same selected items represent the identical outcome regardless of order (like choosing three people for a committee, where the committee is the same regardless of who was "selected first"), it's a combination problem.

Where These Calculations Are Actually Used

Permutations and combinations underpin probability calculations for card games and lotteries, password and PIN security analysis (calculating how many possible combinations an attacker would need to try), scheduling and arrangement problems, and statistical sampling design — any scenario involving counting the number of possible ways to arrange or select from a set of items benefits from correctly applying these formulas rather than attempting to count possibilities manually, which becomes impractical past a small number of items.

Permutations and Combinations With Repetition

Standard permutation and combination formulas assume each item can only be selected once, but some real problems allow repeated selection (like choosing digits for a PIN code, where the same digit can appear multiple times), requiring a modified formula. Recognizing whether a problem allows repetition is an important first step before applying any specific permutation or combination formula.