Student solving n choose r
Count selections where order does not matter.
Choose without repetition and enter n and r.
Gets the exact binomial coefficient and does not confuse it with a permutation.
Combination Calculator
Count unordered selections exactly with or without repetition using arbitrary-precision integer arithmetic and explicit model boundaries.
Specialist tool for Math Science1 documented formula1 guided projectPrivate in your browser
Use this result well
A quick decision brief for this specific tool
Inputs that matter
Whole available-item/type count n, whole selection size r, and an explicit without- or with-repetition model
Output to expect
Exact arbitrary-precision unordered-selection count, digit count, and the actual binomial coefficient used
How it works
Computes C(n,r) multiplicatively without repetition or C(n+r−1,r) when repeated types are allowed, with no floating-point factorial quotient
- Without repetition, r cannot exceed n; with repetition, n represents at least one available type and n+r−1 must stay within the documented bound.
- A combination ignores order. If order matters, items are not distinct, or category limits differ, use a permutation or application-specific counting model.
Choose your path
Built around the job you need to finish
Count unordered selections exactly, with an explicit choice between no repetition and repeated item types.
Planner allowing repeated types
Model selections such as choosing flavors with repeats.
Choose with repetition and enter available types and selection size.
Sees the expanded C(n+r−1,r) formula and exact count.
Analyst checking a large search space
Avoid factorial overflow and rounded counts.
Enter a valid large n and r through the documented bound.
Gets an arbitrary-precision integer plus digit count.
Authoritative checks for this tool
Outputs and checklists are planning aids. Review the linked current authorities and the records, terms, instructions, and requirements that apply to your exact situation before a consequential decision.
- Contemporary Mathematics — Combinations and PermutationsOpenStax, Rice University · Educational definition of combinations as unordered selections and the n-choose-r formula.
- DLMF §26.3 — Lattice Paths: Binomial CoefficientsNational Institute of Standards and Technology · Authoritative binomial-coefficient notation and identities supporting exact multiplicative computation.
- ECMAScript — BigInt ObjectsEcma International · Normative arbitrary-precision integer type used so large combination counts do not overflow or silently round.
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Projects using Combination Calculator
This tool is part of these guided projects. Each project provides step-by-step instructions with checklists and all the tools you need in one place.
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Reference & details
How it works
Updated August 2026
How it works
Updated August 2026Without repetition
For n distinct available items and r selected items where order does not matter, the binomial coefficient gives the count.
C(n,r) = n! / (r!(n−r)!)With repetition
When n represents available types and a type may be selected repeatedly, stars-and-bars converts the problem to a larger binomial coefficient.
C(n+r−1,r)Exact multiplicative computation
The engine multiplies and divides exact integers symmetrically, avoiding huge intermediate factorials and preserving every digit through the documented bound.
Updated: August 2026
Example Scenarios
A student counts ways to choose 3 people from 10 when the order of the selected people does not matter.
A planner counts ways to select a fixed number of items from several types when a type can appear more than once.
An analyst evaluates 100 choose 50 and retains the complete exact coefficient for a downstream model.
Common Mistakes to Avoid
Common Mistakes to Avoid
Counting order when order does not matter
A combination treats the same selected items in different orders as one outcome; use permutations if order changes the outcome.
Choosing the wrong repetition model
Without repetition n counts distinct items; with repetition n counts available types. Confirm the real process before selecting the option.
FAQ
It counts ways to select r items from n distinct available items when order does not matter and items cannot repeat.
There, n is the number of available item types and repeated types are allowed. The count becomes C(n+r−1,r).
Selecting more distinct items than are available is impossible when each item can be used only once.
Yes. A multiplicative BigInt algorithm returns the complete integer without floating-point factorial overflow or rounding.
Use a permutation model when changing the order creates a different outcome. A combination deliberately treats different orders of the same selected items as one selection.
About Combination Calculator
Choose whether repeated item types are allowed, then enter n and r. The calculator returns the complete exact integer, identifies the coefficient used, and rejects impossible or out-of-range models.