Permutation vs Combination: What Is the Difference?
Permutations count arrangements where changing the order creates a different result. Combinations count selections where order does not matter. The quickest way to choose between them is to ask one question: if I rearrange the selected items, do I get a different outcome?
For example, awarding gold, silver and bronze medals uses a permutation because the positions are different. Choosing three people for the same committee uses a combination because the selected group stays the same regardless of selection order.
Permutation and Combination Formulas
Permutation without repetition:
P(n,r) = n! / (n−r)!
Combination without repetition:
C(n,r) = n! / [r!(n−r)!]
Permutation with repetition:
nr
Combination with repetition:
C(n+r−1,r)
Here, n is the number of available items, r is the number selected, and ! means factorial. A factorial multiplies a whole number by every positive whole number below it, so 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1.
How to Use the Calculator
- Enter n, the total number of available items.
- Enter r, the number of items selected or positions filled.
- Choose whether an item can be selected more than once.
- Press Calculate.
- Compare the permutation result, where order matters, with the combination result, where order does not matter.
Worked Examples
Example 1: Permutation — Medal Positions
Ten runners compete for gold, silver and bronze. Because first, second and third place are different positions, order matters.
P(10,3) = 10! / 7! = 10 × 9 × 8 = 720
Answer: There are 720 possible podium arrangements.
Example 2: Combination — Choosing a Committee
A group has 10 people and you need to choose 3 committee members. The order in which the three people are chosen does not create a different committee.
C(10,3) = 10! / (3! × 7!) = 120
Answer: There are 120 possible committees.
With Repetition vs Without Repetition
| Situation | Does order matter? | Can items repeat? | Counting method |
|---|---|---|---|
| Gold, silver and bronze from 10 runners | Yes | No | Permutation: P(10,3) |
| Choose 3 committee members from 10 people | No | No | Combination: C(10,3) |
| Create a 4-digit PIN from 10 digits | Yes | Usually yes | Permutation with repetition: 104 |
| Choose 3 scoops from 5 flavors when flavors may repeat | No | Yes | Combination with repetition: C(7,3) |
Quick decision rule: If rearranging the selected items changes the result, use a permutation. If it does not, use a combination. Then separately decide whether repetition is allowed.
Common Permutation and Combination Mistakes
Using permutations when order does not matter. Selecting A, B and C for the same committee is one combination, not six different selections.
Using combinations when positions are different. Assigning A as president and B as vice president is different from assigning B as president and A as vice president.
Ignoring repetition. A PIN that allows repeated digits follows a different counting rule from arranging distinct digits without replacement.
Entering r greater than n without repetition. You cannot select more distinct items than are available.
How to Check Your Answer
For calculations without repetition, permutations and combinations satisfy:
P(n,r) = C(n,r) × r!
For example, C(10,3) = 120 and 3! = 6, so 120 × 6 = 720 = P(10,3).
You can also verify combinations with the symmetry identity:
C(n,r) = C(n,n−r)
For example, C(10,3) and C(10,7) are both 120.
When Permutations and Combinations Are Useful
These counting methods are used in probability, scheduling, team selection, rankings, codes, games, sampling and many other problems. They are especially useful when you need to count possible outcomes without listing every outcome manually.
For probability questions built from these counts, use the Probability Calculator. You can also browse the full Statistics Calculators collection.
Calculation Methodology and Limits
This calculator validates n and r as non-negative whole numbers within the supported input range. For calculations without repetition, it uses direct multiplicative integer formulas rather than floating-point factorial approximations. For calculations with repetition, it uses nr for ordered selections and C(n+r−1,r) for unordered selections.
The calculator counts possible arrangements or selections; it does not enumerate every arrangement and does not solve arbitrary constraint problems such as adjacency restrictions, forbidden positions or custom inclusion-exclusion cases.
Frequently Asked Questions
nPr counts arrangements where order matters. nCr counts selections where order does not matter. For the same n and r without repetition, nPr equals nCr multiplied by r!.
n is the total number of available items. r is the number of items selected or the number of positions being filled.
Use a permutation when changing the order produces a different outcome. Examples include medal positions, ranked winners, seat assignments and ordered codes.
Use a combination when only the selected group matters. Choosing committee members, lottery numbers or a subset of items are common examples.
“n choose r” is another name for the combination C(n,r), also written as the binomial coefficient. It counts the number of ways to select r items from n items when order does not matter.
Defining 0! as 1 keeps factorial identities and permutation and combination formulas consistent at boundary cases such as C(n,0) = 1 and P(n,0) = 1.
Yes. If each of r ordered positions can independently use any of n choices, the number of ordered selections with repetition is nr.
Yes. When you choose r items from n types, repeats are allowed and order does not matter, the standard count is C(n+r−1,r).
Next step: Working on a probability problem? Use the Probability Calculator, or explore more Statistics Calculators.
