What is the Permutation & Combination Calculator?
This permutation and combination calculator counts selections from n distinct objects or types. A permutation treats different orders as different outcomes; a combination ignores order. Choose whether a type can be selected repeatedly, then enter the number available n and the selection size r.
The answer is an exact integer count, not a probability. Your Probability Calculator uses favorable and total equally likely outcomes; this tool can help count such outcomes only when your counting assumptions match that probability model. It does not assume that every outcome is equally likely.
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Optional scientific calculators (paid link) can help with small manual checks, although many physical calculators lose exactness for large integers. No purchase is required.
How to use the nPr and nCr calculator with steps
- Choose permutations when order matters or combinations when order does not matter.
- Choose without repetition or repetition allowed.
- Enter whole numbers n and r from 0 to 10,000.
- Without repetition, keep r no greater than n.
- Calculate and compare the selected count, alternative count and input-based working.
For example, choosing three people for three different roles is ordered, while selecting a three-person committee is unordered. Repetition allowed means the same type may fill more than one position or be chosen more than once. It does not mean that all physical objects become identical.
Optional probability workbooks (paid link) can provide practice distinguishing these cases. This calculator returns counts, not the full list of arrangements or selections.
Permutation and combination formulas with and without repetition
Without repetition: nCr = n! / [r! × (n − r)!]
Ordered selections with repetition: n^r
Unordered selections with repetition: C(n + r − 1, r), for n ≥ 1
Factorial means a descending product of positive integers, with 0! = 1. For permutations without repetition, the algorithm multiplies only the needed descending factors. For combinations, it uses symmetry and a multiplicative integer recurrence instead of building enormous factorials and dividing approximate decimals.
With repetition, ordered selections have n choices at each position. Unordered selections use the combinations-with-repetition formula, sometimes called the stars-and-bars count. These formulas do not directly count circular arrangements or permutations of a collection containing specified numbers of indistinguishable objects.
Permutation vs combination examples
From six distinct objects, choosing two in order gives 6P2 = 6 × 5 = 30. Choosing two without order gives 6C2 = 30/2! = 15, because each unordered pair appears in two different orders in the permutation count.
From three distinct types, filling four ordered positions with repetition gives 3^4 = 81. Selecting an unordered group of four with repetition gives C(3 + 4 − 1, 4) = C(6, 4) = 15. These are different tasks even though both allow a type to be reused.
Selecting zero items has exactly one outcome: the empty selection. That includes n = 0, r = 0. With repetition allowed, no nonempty selection is possible from zero types, so n = 0 and r > 0 gives zero. Without repetition, r > n is rejected.
Understand exact counts and the alternative result
The large answer corresponds to the selected order and repetition settings. The alternative card applies the other order rule to the same n, r and repetition choice. Use it to understand the role of order, not to add two counts for one event.
Exact integer arithmetic avoids rounding the count to a floating-point approximation. The selected answer is limited to 2,000 digits. If the selected count fits but the alternative is larger, the alternative card explains its display limit instead of blocking the valid selected answer.
Optional combinatorics reference books (paid link) can help with the underlying counting model. For a separate sequence of numeric terms rather than selection counts, use the existing Arithmetic & Geometric Sequence Calculator.
Supported counting problems and limitations
Inputs must be nonnegative whole numbers from 0 to 10,000. Without repetition, r ≤ n. Empty selections are handled by counting conventions, so an ordered selection with n = r = 0 counts once; this does not define 0^0 for every other mathematical setting.
No letters are parsed, no arrangements are generated, and no circular seating, constrained positions, identical-object multiplicities or conditional probability is solved automatically. Determine the correct model before using a count in another calculation. Large inputs can exceed the exact-output bound even when they are valid integers.
Optional graph paper notebooks (paid link) can help list small cases and check whether order changes the outcome. The existing Scientific Calculator remains available for general arithmetic and factorials; this tool supplies the distinct n/r counting workflow.
Permutation & Combination Calculator FAQs
A permutation counts ordered selections; a combination counts selections without regard to order. Choose the rule that matches the actual task.
Yes. Ordered repetition uses n^r. Unordered repetition uses C(n+r−1,r) for n≥1, with empty-pool cases handled explicitly.
n is the number of distinct available objects or types; r is the selection size or number of ordered positions. Both must be whole numbers from 0 to 10,000.
There is exactly one empty selection. This includes n=0 and r=0 under this counting convention.
Yes. Integer arithmetic is exact within the 2,000-digit selected-output limit. An oversized alternative count is labeled separately.
It returns counts only. It does not assign probabilities, generate lists, parse letters or solve constrained/circular/identical-object arrangements.
Sources and methodology
Independently implemented exact arithmetic, reviewed October 25, 2026. The references explain the underlying mathematics; the linked organizations do not endorse this tool.