Mathematics
Calculate permutations and combinations for selecting or arranging items from a set.
Combinations (nCr)
120
Ways to select 3 items from 10 where order does not matter.
Combinations: order does not matter (selecting a team). Permutations: order does matter (arranging in a line).
Permutations and combinations are fundamental counting principles. A permutation (nPr) counts the number of ways to arrange r items chosen from n total items, where order matters—like seating people in specific chairs. A combination (nCr) counts the number of ways to select r items from n items where order does not matter—like choosing team members. Both use factorial notation: nPr = n!/(n-r)! and nCr = n!/(r!(n-r)!).
Enter your total number of items (n) and the number of items you're selecting or arranging (r). The calculator verifies that r is not greater than n, then computes both formulas: nPr by dividing n! by (n-r)!, and nCr by dividing n! by the product of r! and (n-r)!. Factorials are computed by multiplying all positive integers up to that number.
nPr = n! / (n-r)! | nCr = n! / (r! × (n-r)!)n! (n factorial) = n × (n-1) × (n-2) × ... × 1. For permutations, divide n! by (n-r)! to account for arrangements. For combinations, divide by both r! and (n-r)! to eliminate duplicate selections where order doesn't matter. The extra r! in the denominator of nCr accounts for the fact that each selection can be arranged r! ways, but those arrangements are counted as identical.
| Input | Result | Notes |
|---|---|---|
| n = 5, r = 3 | nCr = 10 combinations, nPr = 60 permutations | Selecting 3 from 5 items: 10 different groups (ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE). Arranging 3 from 5: 60 different sequences because each group can be ordered 6 ways (3! = 6). |
| n = 10, r = 3 | nCr = 120 combinations, nPr = 720 permutations | Selecting 3 lottery numbers from 10: 120 ways. Arranging 3 runners in a race from 10 competitors: 720 ways (first place has 10 choices, second has 9, third has 8). |
| n = 7, r = 2 | nCr = 21 combinations, nPr = 42 permutations | Choosing 2 toppings from 7 pizza toppings: 21 combinations. Assigning first and second prize from 7 contestants: 42 permutations. |
In a permutation, order matters—ABC is different from BAC. In a combination, order does not matter—ABC and BAC are the same selection. Use permutations for rankings, sequences, or arrangements; use combinations for teams, committees, or unordered selections.
Because combinations count each unique selection once, while permutations count all possible orderings of that selection. If you have 3 items, there are 3! = 6 ways to arrange them, so nCr is typically r! times smaller than nPr.
If r equals n, you are selecting or arranging all items. nCr = 1 (only one way to select everything), and nPr = n! (all possible orderings of all items).
Yes. Selecting zero items from n items (nC0) always equals 1 because there is exactly one way to select nothing. Similarly, arranging zero items (nP0) equals 1.
If a lottery asks you to pick 6 numbers from 49, that is a combination: 49C6 ≈ 10 million. Your odds of winning with one ticket are 1 in 10 million. Use combinations because the order you pick the numbers does not affect the outcome.
Factorials grow very fast. Even 20! exceeds 2 quintillion. For very large values, approximations (Stirling's formula) may be used, but this calculator computes exact values up to practical limits.