Permutation And Combination Calculator Online

This Permutation and Combination Calculator helps you count how many possible outcomes exist in a selection, arrangement, ranking, draw, password, team, or repeated-choice problem. Enter the total number of items, choose how many are selected or arranged, and the tool shows the exact result with the formula and plain-English explanation.

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What n and r mean

n is the total number of available items. r is how many items are chosen, arranged, or placed into positions.

Example: if there are 10 people and you choose 3 of them, then n = 10 and r = 3. The correct formula depends on whether order matters and whether the same item can be chosen more than once.

Vizualisation of Permutation And Combination

Permutation: order matters

Use a permutation when different orders count as different outcomes.

Example: choosing 1st, 2nd, and 3rd place winners from 10 people is a permutation. Alice, Bob, Chris is different from Bob, Alice, Chris because the ranking changed.

The formula is:

nPr = n! / (n – r)!

Common uses:

• ranking winners
• assigning seats
• arranging selected items
• ordering finalists
• creating codes where repeated characters are not allowed

Combination: order does not matter

Use a combination when the group matters, but the order does not.

Example: choosing 3 people from 10 for a committee is a combination. Alice, Bob, Chris is the same group as Chris, Bob, Alice.

The formula is:

nCr = n! / (r! × (n – r)!)

Common uses:

• choosing a team
• forming a committee
• lottery-style number picks
• selecting toppings when order does not matter
• choosing survey winners without ranking them

Permutation with repetition

Use permutation with repetition when order matters and the same item can be used again.

Example: a 4-digit PIN using digits 0 through 9 has 10 choices for each position. The same digit can repeat, so 1111 is allowed. The result is 10 × 10 × 10 × 10 = 10,000.

The formula is:

n^r

Common uses:

• PIN codes
• password patterns
• license-style codes
• ordered choices where repeats are allowed

Combination with repetition

Use combination with repetition when order does not matter and the same type can be chosen more than once.

Example: choosing 3 scoops of ice cream from 10 flavors can allow repeats. Vanilla, vanilla, chocolate is a different selection from vanilla, chocolate, strawberry, but vanilla, vanilla, chocolate is the same no matter how you list it.

The formula is:

C(n + r – 1, r)

Common uses:

• choosing repeated menu items
• selecting scoops, toppings, or product types
• counting grouped choices where repeats are allowed
• multi-select problems where order does not matter

Factorial

A factorial counts how many ways all distinct items can be arranged.

For example, 5! means:

5 × 4 × 3 × 2 × 1 = 120

Factorials are used inside permutation and combination formulas. The calculator also supports the special case 0! = 1.

How to choose the right mode

Ask two questions:

• Does order matter? If yes, use a permutation. If no, use a combination.
• Can the same item be chosen more than once? If yes, use a repetition mode.

Why the answers can be very different

The same n and r can produce very different results depending on the mode. For n = 10 and r = 3, a permutation gives 720, while a combination gives 120. The permutation is larger because it counts different orders separately.

Repetition can also increase the number of outcomes. A 4-position code using 10 digits with repetition has 10,000 possible outcomes because each position can use any digit again.

Real examples

Ranking winners

If 12 people enter a contest and you need 1st, 2nd, and 3rd place, use permutation. The order matters because each prize position is different.

Choosing a committee

If 12 people are available and you choose 3 for a committee, use combination. The same 3 people form the same committee no matter how their names are ordered.

PIN codes and passwords

If a code has 4 positions and each position can use any of 10 digits, use permutation with repetition. Order matters, and digits can repeat.

Lottery-style selections

If you choose numbers and the draw order does not matter, use combination. A lottery ticket with 3, 8, 15, 22, 31, 44 is the same selection no matter which of those numbers was drawn first.

Menu choices with repeats

If you choose 3 items from several types and repeats are allowed, use combination with repetition. This works when order does not matter but duplicate choices are allowed.

What the comparison table shows

The calculator compares four methods for the same n and r: permutation without repetition, combination without repetition, permutation with repetition, and combination with repetition. This helps you see whether you picked the correct formula.

If one method says “Not valid,” it usually means the situation is impossible under that rule. For example, you cannot choose 8 distinct items from only 5 items without repetition.

Input tips

• Use whole numbers only.
• Use n for the total number of available items.
• Use r for the number chosen, arranged, or placed into positions.
• If repetition is not allowed, r cannot be greater than n.
• If repetition is allowed, r can be greater than n in many problems.
• Use the example type dropdown if you are not sure which mode fits your problem.

Common mistakes

• Using combination for rankings: rankings need permutations because order matters.
• Using permutation for teams: teams usually need combinations because order does not matter.
• Forgetting repetition: passwords, PINs, and repeated menu choices often allow repeats.
• Using r greater than n without repetition: you cannot choose more unique items than exist.
• Confusing factorial with nPr: n! arranges all n items, while nPr arranges only r of them.

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