Permutation vs Combination Guide
Permutation and combination problems often use the same n and r values but count different outcomes. The key question is whether changing the order creates a different result.
Method
- Identify n, the total number of available items, and r, the number selected.
- Ask whether order matters: use a permutation when ABC and ACB are different outcomes, or a combination when they are the same selection.
- Decide whether an item can be reused. Without repetition, r cannot exceed n.
- Use the calculator’s exact integer result rather than rounding large factorial expressions.
Worked example
Choosing president, vice-president and secretary from 10 people is an ordered permutation: 10P3 = 720.
Choosing a 3-person committee from the same 10 people is an unordered combination: 10C3 = 120.
Notes
- With repetition, ordered selections use n^r and unordered selections use C(n+r−1,r).
- The calculator counts mathematical outcomes; real constraints in a word problem can require a different model.
- n and r are bounded to 200 so exact BigInt results remain practical in a browser.
The first question is whether order matters
A permutation treats different orders as different outcomes, while a combination treats selections with the same members as one outcome regardless of order. Choosing a president, vice president and secretary from a group is ordered; choosing three committee members is not.
Writing a small example can prevent formula mistakes. If AB and BA represent different valid outcomes, the problem is order-sensitive.
Check whether repetition is allowed
Many textbook formulas assume selection without replacement, but passwords, repeated draws with replacement and sequence problems can allow the same option more than once. Repetition changes the counting rule.
Define the sample space in words before calculating. A correct factorial expression for the wrong model still answers the wrong question.
Five people, three roles: choosing members is not assigning roles
A small business has five volunteers and needs a three-person review committee. If all members have the same status, the question is how many groups of three can be selected. With no repetition, the count is C(5,3) = 5! ÷ (3! × 2!) = 10. The committee {A, B, C} is the same committee as {C, A, B}. A combination is the appropriate model because the order in which names are listed does not change the result.
If the business instead needs a chair, a reviewer and a note-taker selected from those five people, the positions are distinct. There are five choices for chair, four remaining choices for reviewer and three for note-taker: 5 × 4 × 3 = 60 assignments, or P(5,3). Every three-person committee can fill those three roles in 3! = 6 ways, so 10 × 6 = 60. Enter the same n = 5 and r = 3 into the linked calculator and switch models to see why the answers differ.
A repeated option changes the count before any factorial is used
For a three-character numeric code using digits 0 through 9, if leading zeros and repeated digits are permitted, each position has ten choices and the count is 10 × 10 × 10 = 1,000 distinct strings. If digits cannot repeat but a leading zero is still permitted, the count is 10 × 9 × 8 = 720. The same three positions and ten symbols give different results because the replacement rule changed. These are counting examples, not recommendations for secure authentication codes.
Before choosing a formula, list three assumptions: whether order creates distinct outcomes, whether an item can appear more than once and whether any position has special restrictions. A no-leading-zero rule, for example, would require a different first-position count. If the options have unequal probabilities, counting distinct outcomes alone is not enough to calculate event probabilities. State those restrictions beside the result so another person can reproduce it.
Editorial review and limitations
Continue with the related tools
Use the linked tools for your own inputs and scenarios.
- Binomial Probability Calculator — Calculate exact, at-most and at-least binomial probabilities from trials, successes and a fixed success probability.
- Normal Distribution Calculator — Calculate normal-distribution left-tail, right-tail and between-values probabilities, density, z-scores and inverse percentiles.
- Mean Median Mode Calculator — Calculate mean, median, mode, minimum and maximum from a list of numeric values.