Combinations (nCr) & Permutations (nPr) Calculator
Calculate the number of possible combinations C(n,r) and permutations P(n,r) when selecting r items from a set of n.
About the Combinations (nCr) & Permutations (nPr) Calculator
In combinatorics and probability, Combinations (nCr) count how many ways you can select r items from a group of n items when order does not matter. Permutations (nPr) count how many ways you can arrange r items from n when order does matter.
Our calculator computes both C(n, r) and P(n, r) simultaneously using exact factorials.
Combinations (nCr) & Permutations (nPr) Formulas
When selecting r items from a set of n distinct items (where 0 ≤ r ≤ n):
C(n, r) = n! / ( r! × (n − r)! )P(n, r) = n! / (n − r)!How to Calculate nCr and nPr Step-by-Step
- Calculate Permutations P(n, r) First: Multiply
rdescending factors starting fromn, which equalsn! / (n − r)!. - Divide by r! for Combinations C(n, r): Since order does not matter in combinations, divide
P(n, r)byr!to remove duplicate orderings.
Worked Example: To find 5 choose 2, first find P(5, 2) = 5 × 4 = 20. Then divide by 2! = 2 to get C(5, 2) = 10 combinations.
Frequently Asked Questions (FAQ)
When should I use Combinations vs. Permutations?
Use Permutations when the order of selection creates a different outcome (such as a podium finish 1st/2nd/3rd or a PIN code). Use Combinations when order does not matter (such as choosing a 3-person committee or lottery numbers).
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