Possible Combinations Calculator

About the Possible Combinations Calculator

How many outcomes, arrangements, or selections are possible? The answer depends on two questions: does order matter, and is repetition allowed? Those two choices define the four standard counting formulas in introductory combinatorics.

This calculator brings the four cases together in one place: permutations without repetition, permutations with repetition, combinations without repetition, and combinations with repetition. Enter n (pool size) and r (selection size), then compare the formulas side by side if you are unsure which model fits your problem.

That makes the page useful for lottery odds, password spaces, seating orders, team selection, and quick classroom checks when the main challenge is identifying the right formula before doing the arithmetic.

When This Page Helps

The hardest part of many counting problems is not the calculation itself but recognizing which assumptions apply. Showing all four formulas together makes it easier to spot whether your scenario is really a combination, a permutation, or a with-repetition case before you carry the result into a probability or odds calculation.

How to Use the Inputs

1. Ask: does order matter? (yes → permutation, no → combination)
2. Ask: can items repeat? (yes → with repetition, no → without)
3. Select the appropriate counting type.
4. Enter n (total items/types) and r (items to choose/positions).
5. Or click a preset for common scenarios.
6. Review your result and compare with all four formulas.
7. Use the decision guide and examples table for reference.

Example Calculation

Tips & Best Practices

• If you're unsure which formula to use, start with the decision matrix: does order matter? Can items repeat?
• The four formulas always satisfy: n^r ≥ P(n,r) ≥ C(n,r) and C(n+r−1,r) ≥ C(n,r).
• P(n,r) = C(n,r) × r! — permutations are combinations times arrangement factor.
• For passwords and PINs, use n^r (ordered with repetition).
• For lottery draws, use C(n,r) (unordered without repetition).
• When in doubt, enumerate a tiny example (n=3, r=2) to verify which formula matches.

The Four Counting Paradigms

Every counting problem in elementary combinatorics falls into one of four categories based on two binary choices: ordered vs unordered, and with vs without repetition. Understanding which paradigm applies is the crucial first step. The decision matrix in this calculator encodes this logic visually.

From Counting to Probability

Probability = favorable outcomes / total outcomes. Each counting formula directly yields the denominator. If you want the probability of a specific event (drawing a royal flush in poker), count favorable outcomes (4) and divide by total outcomes (C(52,5) = 2,598,960), giving P ≈ 1.54 × 10⁻⁶.

Advanced Counting Techniques

Beyond the four basic formulas, advanced counting uses: inclusion-exclusion (for overlapping constraints), the pigeonhole principle (existence proofs), generating functions (for complex recurrences), Burnside's lemma (for symmetry), and bijective proofs (mapping to known counts). These build on the four fundamentals covered here.

Frequently Asked Questions

• Which formula should I use?

  Ask two questions: (1) Does order matter? If choosing items A, B and B, A are different, order matters → permutation. (2) Can items be reused? If yes → with repetition. This gives you exactly one of the four formulas.

• What is the relationship between the four formulas?

  n^r (ordered, repeated) ≥ P(n,r) (ordered, no repeat) ≥ C(n,r) (unordered, no repeat). Adding repetition increases counts; adding order increases counts. C(n+r−1,r) (unordered, repeated) is between C(n,r) and n^r.

• What if n = r?

  P(n,n) = n! (all permutations of n items). C(n,n) = 1 (only one way to choose all items). n^n (permutations with repetition). C(2n−1,n) for combinations with repetition.

• How do I count outcomes with constraints?

  For constrained counting (at least one digit, no consecutive letters, etc.), use inclusion-exclusion: count unconstrained outcomes, then subtract forbidden ones. Or break into cases and sum.

• What is the multiplication principle?

  If stage 1 has n₁ outcomes and stage 2 has n₂ outcomes, the total is n₁ × n₂. This generalizes to any number of stages and underlies all four counting formulas. It's the most fundamental rule in combinatorics.

• Can these formulas handle very large numbers?

  The formulas work for any n and r, but the results can be astronomically large. C(100,50) ≈ 10²⁹. For practical purposes, logarithmic representations or scientific notation are used when results exceed standard number ranges.