Compute both permutations (nPr) and combinations (nCr) for chosen n and r values with formula display.
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Permutations & Combinations
From 10 items choosing 3: there are 720 permutations (order matters) and 120 combinations (order doesn’t matter).
Formulas
P(10,3) = 10! / 7! = 720
C(10,3) = 10! / (3! × 7!) = 120
Also in Probability
Combinatorics
A permutation and combination calculator finds both the number of ordered arrangements (nPr) and unordered selections (nCr) for a given n and r. It displays the formulas, factorial breakdowns, and lets you compare the two results side by side.
The permutation formula P(n,r) = n! / (n−r)! counts ordered arrangements. It multiplies n × (n−1) × … for r terms. For example, P(10,3) = 10 × 9 × 8 = 720 different sequences of 3 items from 10.
Use permutations when order matters: ranking finishers in a race, assigning distinct roles, or creating codes where digit position matters.
P(n, r) = n! / (n − r)!
The number of ordered arrangements of r items from n items.
The combination formula C(n,r) = n! / (r! × (n−r)!) removes ordering by dividing the permutation count by r!. Each group of r items can be rearranged in r! ways, so C(n,r) is always less than or equal to P(n,r).
For C(10,3): 720 permutations / 6 orderings = 120 combinations. The classic example is poker hands: C(52,5) = 2,598,960 distinct 5-card hands from a 52-card deck.
C(n, r) = n! / (r! × (n − r)!)
The number of unordered selections of r items from n items.
Frequently asked questions
Use permutations when the order of selection matters (e.g., race placements, PIN codes). Use combinations when only the group matters, not the arrangement (e.g., lottery numbers, team selection).
Because every combination of r items corresponds to r! different permutations. Dividing nPr by r! gives nCr, so nPr = nCr × r!. They are equal only when r is 0 or 1.
Related
These related calculators come from the same leaf category, nearby sibling categories, or the same top-level topic.
Calculate the number of ways to choose r items from n items without regard to order using the nCr formula.
Calculate the number of ordered arrangements of r items chosen from n items using the nPr formula.
Compute probability for single events, complements, and independent compound events (AND/OR) with decimal and percentage outputs.
Calculate the probability of rolling a specific sum with one or more dice of any polyhedral type.
