Choosing the right formula: order and repetition
Four distinct counting formulas apply depending on two independent yes/no questions: does order matter, and is repetition allowed?
| Order matters? | Repetition allowed? | Formula | Example scenario |
|---|---|---|---|
| Yes | No | nPr = n! ÷ (n − r)! | Awarding 1st/2nd/3rd place among 7 runners |
| Yes | Yes | nʳ | Creating a 3-digit PIN from digits 0–6, repeats allowed |
| No | No | nCr = n! ÷ (r!(n−r)!) | Choosing a 3-person committee from 7 people |
| No | Yes | C(n+r−1, r) | Choosing 3 scoops from 7 ice cream flavors, repeats allowed |
- nCr is always less than or equal to nPr for the same n and r (specifically nCr = nPr ÷ r!), because combinations collapse every group of r! orderings of the same items into a single count.
- When r = 0, both nPr and nCr equal 1 by convention — there is exactly one way to select zero items (the empty selection).
- When r = n (no repetition), nPr = n! and nCr = 1 — there is only one way to 'choose' all n items as an unordered set, but n! ways to arrange all of them in order.
What are permutations and combinations?
A permutation is an arrangement of items in which the order matters. The number of permutations of r items selected from a set of n distinct items (without repetition) is written nPr and counts arrangements such as 'first place, second place, third place' where swapping two selected items produces a different, distinct outcome.
A combination is a selection of items in which the order does not matter. The number of combinations of r items chosen from n distinct items is written nCr (also called 'n choose r' or the binomial coefficient) and counts groupings such as a committee or a hand of cards, where the same set of items in a different order is considered the same outcome.
The key distinguishing question is: does rearranging the same selected items produce a different result? If yes (e.g. assigning gold/silver/bronze medals, or arranging books on a shelf), use permutations. If no (e.g. choosing a 3-person committee, or picking a hand of cards), use combinations. This calculator also reports the 'with repetition' variants, which apply when the same item can be selected more than once (e.g. choosing digits for a PIN, or scoops of ice cream flavors where repeats are allowed).
How to use this permutation and combination calculator
- Enter n — the total number of distinct items available to choose from.
- Enter r — the number of items to be selected or arranged. r must be less than or equal to n for the no-repetition results (repetition-allowed results permit any r ≥ 0).
- Read the permutations (nPr) if the order of selection matters and repetition is not allowed, or combinations (nCr) if order does not matter and repetition is not allowed.
- For scenarios allowing the same item to be chosen more than once, use the 'with repetition' results instead of the standard nPr/nCr values.
The permutation and combination formulas
The number of permutations of r items chosen from n, without repetition, is nPr = n! ÷ (n − r)!, equivalently the product of r consecutive descending integers starting at n. Worked example with n = 7, r = 3: 7P3 = 7 × 6 × 5 = 210 — the number of ways to award distinct 1st, 2nd and 3rd place prizes among 7 competitors.
The number of combinations of r items chosen from n, without repetition, is nCr = n! ÷ (r! × (n − r)!) = nPr ÷ r!. This divides out the r! ways of ordering each selected group, since combinations do not distinguish order. Worked example with n = 7, r = 3: 7C3 = 210 ÷ 3! = 210 ÷ 6 = 35 — the number of distinct 3-person committees that can be formed from 7 people.
Permutations with repetition allowed (each of the r positions can independently be any of the n items) is simply nʳ. For n = 7, r = 3: 7³ = 343 — for example, the number of distinct 3-digit codes using digits 0–6 if repeats are allowed. Combinations with repetition allowed uses the 'stars and bars' formula C(n + r − 1, r). For n = 7, r = 3: C(9, 3) = 84 — for example, the number of ways to choose 3 scoops of ice cream from 7 flavors when repeated flavors are allowed and order doesn't matter.
Common mistakes
- Using the permutation formula when order does not actually matter (or vice versa) — check whether swapping two selected items changes the outcome before choosing a formula.
- Forgetting to check whether repetition is allowed — 'choosing a PIN' (digits can repeat) needs a different formula from 'dealing cards from a deck' (no repetition, cards do not return to the deck).
- Applying nPr or nCr with r greater than n when repetition is not allowed — this is undefined, since you cannot select more distinct items than exist in the set without repeating one.
- Confusing 'combinations with repetition' (the stars-and-bars formula) with plain nCr — these give very different results and apply to different scenarios (repeats allowed vs. not).
Questions fréquentes
What is the difference between a permutation and a combination?
A permutation counts arrangements where order matters (nPr = n! ÷ (n−r)!) — for example, ranking 1st, 2nd and 3rd place. A combination counts selections where order does not matter (nCr = n! ÷ (r!(n−r)!)) — for example, choosing a committee. For n = 7 and r = 3: 7P3 = 210 (ordered), but 7C3 = 35 (unordered), because each group of 3 people can be arranged in 3! = 6 different orders, and 210 ÷ 6 = 35.
How do you calculate nCr (n choose r)?
Use the formula nCr = n! ÷ (r! × (n − r)!). For n = 7, r = 3: 7C3 = 7! ÷ (3! × 4!) = 5040 ÷ (6 × 24) = 5040 ÷ 144 = 35. Equivalently, compute nPr first (7 × 6 × 5 = 210) and divide by r! (3! = 6): 210 ÷ 6 = 35.
How do you calculate nPr?
Use the formula nPr = n! ÷ (n − r)!, which simplifies to the product of r consecutive descending integers starting from n. For n = 7, r = 3: 7P3 = 7 × 6 × 5 = 210.
When should I allow repetition in a permutation or combination problem?
Allow repetition when the same item can be selected more than once in a single outcome — for example, digits in a PIN code, or ice cream flavors when a customer can choose the same flavor for multiple scoops. Do not allow repetition when each item can only be used once — for example, dealing distinct playing cards, or assigning distinct people to roles.
What is nPr when r equals n?
When r = n (arranging all n items with no repetition), nPr simplifies to n! ÷ (n − n)! = n! ÷ 0! = n! ÷ 1 = n!. This represents the total number of ways to arrange all n distinct items in a row — for example, 5 items can be fully arranged in 5! = 120 ways.
What does 'stars and bars' mean for combinations with repetition?
'Stars and bars' is the standard combinatorial technique for counting combinations with repetition allowed, giving the formula C(n + r − 1, r). It works by representing the r chosen items as 'stars' separated by 'bars' marking the boundaries between the n categories, then counting the arrangements of stars and bars. For n = 7, r = 3: C(9, 3) = 84.
Références
- Rosen KH. Discrete Mathematics and Its Applications. 8th ed. McGraw-Hill, 2018. (Permutations, combinations, and the stars-and-bars method.)
- NIST Digital Library of Mathematical Functions (DLMF), §26.1–26.3 Combinatorial Analysis. dlmf.nist.gov.
- Feller W. An Introduction to Probability Theory and Its Applications, Vol. 1. 3rd ed. Wiley, 1968. (Classic treatment of permutations and combinations.)