What Is a Combinations Calculator?

A combinations calculator computes C(n, r) — the number of ways to choose r items from a group of n items without regard to the order of selection. Also called "n choose r" and written nCr or C(n,r), this is one of the most-used formulas in probability, statistics, card games, lottery analysis, genetics, and computer science. This calculator simultaneously computes both combinations and permutations so you can compare the two at a glance.

For example, choosing 3 members for a committee from 10 candidates: the order in which they are chosen doesn't matter, so we use combinations. C(10, 3) = 120 means there are exactly 120 distinct possible committees.

How Combinations Are Calculated

The formula is: C(n, r) = n! / (r! × (n − r)!)

For C(10, 3): C(10,3) = 10! / (3! × 7!) = 3,628,800 / (6 × 5,040) = 3,628,800 / 30,240 = 120.

The key distinction from permutations: P(n, r) = n! / (n − r)! = 720 for P(10, 3). The relationship is C(n,r) = P(n,r) / r!, which means permutations count each combination r! times (once for each ordering of the r items).

Combinations vs Permutations

Use combinations when order does NOT matter: Lottery numbers (1-2-3-4-5 is the same as 5-4-3-2-1), committee selection, poker hands (the order of dealing doesn't matter), choosing pizza toppings, selecting items from a menu.

Use permutations when order DOES matter: Race rankings (1st, 2nd, 3rd are different), PIN codes, arranging books on a shelf, selecting officers from a club (President ≠ Vice President), anagrams.

The ratio P/C = r! tells you how many times more permutations there are than combinations. For r = 3: P/C = 6. So for every 1 combination, there are 6 ordered permutations.

Applications of Combinations

Lottery probability: Powerball selects 5 numbers from 69. The number of possible tickets is C(69, 5) = 11,238,513. With the additional Powerball from 26, total combinations = 292,201,338. Your probability of winning: 1 in 292 million.

Poker hands: A 5-card hand from a 52-card deck: C(52, 5) = 2,598,960 possible hands. The probability of a royal flush is C(4,1)/C(52,5) = 4/2,598,960 ≈ 1 in 649,740.

Genetics: The number of ways to inherit r traits from n possible traits — genetic diversity calculations use combinations extensively in population genetics.

Pascal's Triangle and Combinations

Pascal's Triangle is a triangular array where entry C(n, r) appears at row n, position r (both 0-indexed). Each entry equals the sum of the two entries directly above it: C(n,r) = C(n−1,r−1) + C(n−1,r). This recursive property makes Pascal's Triangle useful for expanding binomial expressions: (a+b)^n = Σ C(n,k) × a^(n-k) × b^k.

Tips for This Calculator

Enter n (total items) and r (items to choose) as non-negative integers with r ≤ n. The maximum supported is n = 20 due to integer overflow at larger factorials. For very large values, use logarithms or Stirling's approximation. Pair with the permutations calculator for order-sensitive counting, or the probability calculator for full probability distributions.