Permutation & Combination Calculator

Live P(n,r) vs C(n,r) board with an identity peel that shows why permutations equal combinations times r!, plus a work strip and r-ladder on fixed n.

Educational combinatorics literacy only. Not a probability engine, not a full CAS for huge n, and not an exam oracle. Match the formula to whether order matters in your problem.

By Ahmet C. Toplutaş·Site owner & editor · Math tools

Last updated May 21, 2026 · Reviewed May 21, 2026 · Planning estimates only - not tax, medical, or investment advice

Order matters vs order does not

Combinatorics homework usually asks one of two questions. Permutations count ordered arrangements: who finishes 1st / 2nd / 3rd, which PIN digits in which sequence. Combinations count unordered selections: which four people join a committee, which six lottery numbers, ignoring draw order.

This page’s framing is a P(n,r) vs C(n,r) twin, with an identity peel that shows P = C × r!, a work strip for the falling product, and an r-ladder on fixed n. It is not the mean/median/mode three-center board, not fraction simplify work, and not a roofing pitch ladder.

Decision framing: “If I reshuffle the same r items, does that count as a new outcome? Yes → permutation. No → combination.”

Educational combinatorics literacy only. Not a probability model, not a full computer-algebra system for huge n, and not an exam oracle. Confirm methods with your course when grades are on the line. See our disclaimers.

Identity peel: P = C × r!

Every unordered group of r items can be lined up in r! different orders. That is why permutations outnumber combinations by exactly the factorial of r (when C > 0). The peel prints r!, the order multiplier, and a pass/fail identity check so the gap is not a mystery.

Load Race podium (n=8, r=3). C is smaller; P is C × 3! = C × 6. Flip to Committee and watch the multiplier become 4!.

Work strip

P is the falling product n × (n−1) × … × (n−r+1). C uses the multiplicative binomial rule that cancels factorials early so larger classroom values stay exact (BigInt under the hood, n capped for this sketch).

r-ladder on fixed n

Keep the set size fixed and scan several r values. Tap a ladder row to set r without retyping n. Growth in r is the product beat: small r changes can explode P much faster than C.

What the panel actually computes

Standard counting formulas with the linking identity.

P(n,r) = n! / (n − r)! = n × (n−1) × … × (n−r+1) C(n,r) = n! / (r! (n − r)!) P(n,r) = C(n,r) × r! P(n,0) = C(n,0) = 1

Where this hands off

After counting arrangements or selections, hand off for other math literacy tools. Probabilities need a sample-space story on top of these counts.

Centers of a list: mean / median / mode
Percent jobs: percentage calculator
Fraction arithmetic: fraction calculator
Linear algebra sketch: matrix calculator

Worked scenarios

Scenario A: Race podium. 8 runners, top 3 places. Order matters → P(8,3). C(8,3) is the unordered trio count; identity says P = C × 6.

Tap r-ladder rows from r=1 to r=3 on n=8. Watch P pull away from C as r! grows.

Scenario B: Committee. 12 people, choose 4. Order does not matter → trust C. P is the same people seated in 4! labeled chairs.

Scenario C: Lock digits. 10 digits, length-4 code with no repeat in this sketch (standard P). Real locks may allow repeats; that is a different model.

Scenario D: Lottery sketch. 49 choose 6. Combination territory; the board still prints P so the identity multiplier stays visible.

Common mistakes

Using C when order matters (or P when it does not).
Allowing r > n without noticing the block.
Confusing “with replacement” problems with this no-repeat P/C pair.
Treating the count as a probability without dividing by a sample space.
Expecting this page to behave like mean / median / mode (data summary) or a full CAS for enormous n.
Forgetting P(n,0) = C(n,0) = 1 is a valid edge case.

How to use this panel

Load a preset or enter whole-number n and r.
Read the P vs C twin and the identity peel.
Scan the work strip and tap the r-ladder to explore nearby r on the same n.
Share when you want URL state. Typing does not rewrite the address bar. Match the formula to whether order matters in your word problem.

When not to stop at P and C

Stop here for ordered vs unordered counts. Hand off for centers, percents, fractions, or matrices. For probability, build the sample space explicitly. For huge combinatorial identities, use a computer-algebra system.

Mean / median / mode: three centers.
Percentage: percentage.
Fractions: fractions.

Frequently asked questions

How does this permutation and combination calculator work?

It computes P(n,r) = n!/(n−r)! for ordered arrangements and C(n,r) = n!/(r!(n−r)!) for unordered selections. A twin board shows both, an identity peel checks P = C × r!, and an r-ladder varies r on fixed n.

When do I use a permutation vs a combination?

Use a permutation when order matters (race places, PINs, ranked lists). Use a combination when order does not (committees, lottery tickets, hand of cards). The board always shows both so you can compare.

Why is P(n,r) larger than C(n,r)?

Each unordered group of r items can be arranged in r! different orders. So P(n,r) = C(n,r) × r!. The identity peel shows that multiplier explicitly.

What does the r-ladder show?

It keeps n fixed and lists P and C for several r values, including your active r. That makes growth in r visible without retyping n each time.

Can r be zero?

Yes. There is one way to arrange or choose nothing: P(n,0) = C(n,0) = 1. The empty-pick narrative flags that classroom edge case.

Is this a probability calculator?

No. It counts arrangements and selections. Probabilities need a sample-space model on top of these counts. It is also not a full computer-algebra system for huge n.

How is this different from mean/median/mode or fractions?

Mean/median/mode summarizes a data list. Fractions simplify arithmetic. This page is combinatorics literacy: order vs unordered counts with an identity peel and r-ladder.

Sources

Loading…

Order matters vs order does not

Decision framing: “If I reshuffle the same r items, does that count as a new outcome? Yes → permutation. No → combination.”

Identity peel: P = C × r!

Load Race podium (n=8, r=3). C is smaller; P is C × 3! = C × 6. Flip to Committee and watch the multiplier become 4!.

Where this hands off

After counting arrangements or selections, hand off for other math literacy tools. Probabilities need a sample-space story on top of these counts.

Centers of a list: mean / median / mode
Percent jobs: percentage calculator
Fraction arithmetic: fraction calculator
Linear algebra sketch: matrix calculator

Worked scenarios

Scenario A: Race podium. 8 runners, top 3 places. Order matters → P(8,3). C(8,3) is the unordered trio count; identity says P = C × 6.

Tap r-ladder rows from r=1 to r=3 on n=8. Watch P pull away from C as r! grows.

Scenario B: Committee. 12 people, choose 4. Order does not matter → trust C. P is the same people seated in 4! labeled chairs.

Scenario C: Lock digits. 10 digits, length-4 code with no repeat in this sketch (standard P). Real locks may allow repeats; that is a different model.

Scenario D: Lottery sketch. 49 choose 6. Combination territory; the board still prints P so the identity multiplier stays visible.

Common mistakes

Using C when order matters (or P when it does not).
Allowing r > n without noticing the block.
Confusing “with replacement” problems with this no-repeat P/C pair.
Treating the count as a probability without dividing by a sample space.
Expecting this page to behave like mean / median / mode (data summary) or a full CAS for enormous n.
Forgetting P(n,0) = C(n,0) = 1 is a valid edge case.

Frequently asked questions

How does this permutation and combination calculator work?

When do I use a permutation vs a combination?

Why is P(n,r) larger than C(n,r)?

Each unordered group of r items can be arranged in r! different orders. So P(n,r) = C(n,r) × r!. The identity peel shows that multiplier explicitly.

What does the r-ladder show?

It keeps n fixed and lists P and C for several r values, including your active r. That makes growth in r visible without retyping n each time.

Can r be zero?

Yes. There is one way to arrange or choose nothing: P(n,0) = C(n,0) = 1. The empty-pick narrative flags that classroom edge case.

Is this a probability calculator?

No. It counts arrangements and selections. Probabilities need a sample-space model on top of these counts. It is also not a full computer-algebra system for huge n.

How is this different from mean/median/mode or fractions?

Mean/median/mode summarizes a data list. Fractions simplify arithmetic. This page is combinatorics literacy: order vs unordered counts with an identity peel and r-ladder.

Order matters vs order does not

Identity peel: P = C × r!

Work strip

r-ladder on fixed n

What the panel actually computes

Where this hands off

Worked scenarios

Common mistakes

How to use this panel

When not to stop at P and C

Frequently asked questions

How does this permutation and combination calculator work?

When do I use a permutation vs a combination?

Why is P(n,r) larger than C(n,r)?

What does the r-ladder show?

Can r be zero?

Is this a probability calculator?

How is this different from mean/median/mode or fractions?

Sources

Order matters vs order does not

Identity peel: P = C × r!

Work strip

r-ladder on fixed n

What the panel actually computes

Where this hands off

Worked scenarios

Common mistakes

How to use this panel

When not to stop at P and C

Frequently asked questions

How does this permutation and combination calculator work?

When do I use a permutation vs a combination?

Why is P(n,r) larger than C(n,r)?

What does the r-ladder show?

Can r be zero?

Is this a probability calculator?

How is this different from mean/median/mode or fractions?

Sources

What this number means

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