Combinations calculator

Students, teachers, data scientists, and contest programmers get a deterministic JSON count—no computer-algebra install, no random sampling, and the same pure function in the free browser tool and the prepaid API path—so homework checks, pipeline validation, and interview sandboxes all share one reproducible answer.

What n choose r means and why BigInt matters

Boundary cases are well defined: C(n,0) and C(n,n) are always one, C(n,1) equals n, and any r greater than n or any negative index is rejected as invalid_input rather than inventing an extension. That contract matches standard combinatorics textbooks and programming-contest judges, so you can trust the number against hand calculations for small n and against known tables for larger ones without floating-point drift or silent wraparound.

How to call the API and read the response

Field n is the total number of distinct items—the upper index of the binomial coefficient—and must be an integer from zero through ten thousand. Field r is how many items you choose—the lower index—and must be an integer from zero through n. /(r!( n-r)!) so clients can display or log the identity without hard-coding it. Use it from homework graders, programming-contest validators, interview-prep sandboxes, spreadsheet enrichment jobs, probability demos that need C(n,k) for binomial probabilities, or any workflow that needs a reproducible combinations count without installing multiprecision packages on every machine that runs the pipeline. Send a JSON body with two required non-negative integers. Aliases total and N are accepted for the same meaning. Aliases k, choose, and R map to r. A successful response returns n and r as numbers, count as a decimal string of the exact binomial value, and formula as the string C(n,r)=n! Failed validation—missing n or r, a non-integer, a fraction, a value above the published maximum, a negative number, or r greater than n—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price.

Practical tips, limits, and related intents

When teaching, prefer paired examples: n equals ten and r equals three yields one hundred twenty; n equals five and r equals zero yields one; n equals five and r equals six demonstrates the hard rejection path for r greater than n; and a negative r demonstrates the negative rejection path. Sibling landings may target permutations, factorials, or figurate numbers; this page is specifically for people searching for a combinations calculator, nCr calculator, or binomial coefficient tool. Do not confuse combinations with permutations: if order matters, you need P(n,r) equals n! /(n−r)!, which is a different intent and a different landing. Keep n at most ten thousand so worst-case C(10000,5000) finishes quickly on the Worker and in the browser while still covering contest and teaching sizes far beyond safe Number range. If you need multiprecision coefficients with n in the hundreds of thousands, use a dedicated big-integer stack offline—this lightweight parse capability is built for interactive checks, API glue, and classroom demos. Because the multiplicative formula is deterministic, students can recompute small cases by hand or with a spreadsheet and match the API line by line. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type bad values while learning the contract. Prefer exact string counts over forcing a JavaScript Number so large coefficients never lose low-order digits in JSON transport.