Counting Methods
A fuller guide to permutations and combinations, built around the key decision of whether order matters and how constraints change the counting model.
Key formulas
Order is the first and most important question
Permutations count arrangements where order matters. Combinations count selections where order does not matter. That distinction solves more problems than memorising the symbols alone because it asks what the problem is really treating as different.
If choosing Alice then Ben is considered different from choosing Ben then Alice, you are in permutation territory. If the pair is the same either way around, you are in combination territory.
Factorials explain why the formulas look the way they do
A factorial counts the number of ways to order distinct objects. From there, permutation and combination formulas simply adjust for how many positions are being filled and whether repeated orderings should be treated as duplicates.
Seeing the formulas as structured counting rather than mysterious symbols makes them easier to adapt when constraints are added.
- Permutations of n items taken r at a time: nPr = n! / (n - r)!
- Combinations of n items taken r at a time: nCr = n! / (r!(n - r)!)
Worked examples
Example 1: From 5 people, the number of possible 3-person committees is 5C3 = 10 because committee order does not matter.
Example 2: From the same 5 people, the number of ordered chair-secretary-treasurer assignments is 5P3 = 60 because role order matters.
Example 3: If one person must always be included, count the remaining choices after fixing that condition rather than using the unrestricted formula and hoping to adjust later.
Constraint thinking matters
Real problems often include restrictions: one item must be included, two items cannot appear together, or repetition is allowed. At that point, the standard formulas are still useful, but only after the problem has been reframed carefully.
A good habit is to state the sample space in words before touching the formula. That keeps the counting aligned with the real interpretation.
Common mistakes
- Using combinations when order is actually meaningful.
- Using permutations for unordered groups such as committees or hand selections.
- Ignoring constraints and treating a restricted problem as unrestricted.
- Forgetting that factorial growth makes large answers normal in counting problems.
Apply the topic straight away.
Combinations Calculator
Use the Combinations Calculator to calculate combinations from your own dataset with practical output and sensible validation.
Permutations Calculator
Use the Permutations Calculator to calculate permutations from your own dataset with practical output and sensible validation.