Combinations Calculator

Formula

nCr = factorial(Total items n) ÷ (factorial(Chosen items r) × factorial(Total items n-Chosen items r))

Worked example

With Total items n: 8, Chosen items r: 3, ncr is 56.

Calculation method

The rule is nCr = factorial(Total items n) ÷ (factorial(Chosen items r) × factorial(Total items n-Chosen items r)).

Order does not matter in a combination. The selection {A, B} is counted once regardless of selection order.

For this combinations example, changing total items n from 8 to 9 while keeping the other inputs fixed changes nCr from 56 to 84.

Reading nCr

Order matters for permutations, but not for combinations. This distinction changes the count even when the same items are chosen. In the worked values on this page, total items n 8, chosen items r 3 produce ncr 56.

Practical use

Use these counts to enumerate possible orders or groups in a defined finite set. The specific task here is to find ncr from total items n, chosen items r.

Limits and common mistakes

The total and chosen counts must be nonnegative integers, and chosen items cannot exceed total items. Verify total items n before interpreting nCr; a value measured on another basis would answer a different question.

Why the relationship works

Counting starts with a clear rule about order and repetition. Factorial counts complete arrangements of distinct objects. Permutations count ordered choices of a specified size; combinations count unordered selections. If two choices contain the same objects in a different order, only a permutation counts them separately. The combinations rule uses total items n, chosen items r in nCr = factorial(Total items n) ÷ (factorial(Chosen items r) × factorial(Total items n-Chosen items r)); read each symbol as the field named on this page.

Interpreting another set of values

For combinations, the fields labeled total items n, chosen items r are the known quantities. The calculator solves for ncr using nCr = factorial(Total items n) ÷ (factorial(Chosen items r) × factorial(Total items n-Chosen items r)). Clarify whether the output is an integer count, a selected term, or a sum of terms. The index and the rule about order or repetition determine which mathematical object is being counted. When comparing two calculations, keep their units and input definitions identical.

For a second combinations check, keep the other default inputs fixed and change total items n from 8 to 10. nCr changes from 56 to 120. This is a comparison of two particular cases. Repeat it near the values relevant to your task; do not assume that every formula changes at a constant rate.

Checks and limits

Use whole-number item counts. For selections, chosen items cannot exceed total items. Factorial and choice counts can grow beyond an ordinary numeric display; if every digit matters, use an exact-integer method rather than a rounded decimal value. The reported ncr is 56 in the worked example. Keep its stated unit or counting convention when checking it against another calculation.