Numarly

Combination Calculator (nCr)

Calculate combinations — selection of items (order doesn't matter)

What is it?

A combination counts the number of ways to select r items from a set of n items, where the order of selection does not matter.

Formula

nCr = n! / (r! × (n − r)!)

Formula Explanation

Combination starts from the permutation formula (n! / (n−r)!) and divides by r! to remove the duplicate orderings of the r selected items, since order doesn't distinguish one selection from another here.

Example Calculation

5C2 = 5! / (2! × 3!) = 120 / (2 × 6) = 10 ways to choose 2 items from 5.

How to Use

  1. Enter the total number of items (n).
  2. Enter the number of items being selected (r).
  3. Click Calculate nCr to see the number of possible selections.

Benefits

  • Instantly computes exact selection counts without manual factorial arithmetic.
  • Shows the formula substitution step for learning purposes.
  • Validates inputs to prevent invalid r > n calculations.

Use Cases

  • Choosing a committee or team from a larger group of candidates.
  • Lottery and card game probability calculations.
  • Counting subsets of a given size from a larger set.

What Your Result Means

The result is the exact number of distinct groups of r items possible from a set of n, where the order within each group does not matter — choosing {A, B} is the same as choosing {B, A}.

Tips

  • Use combination when only the group matters, not the arrangement (unlike permutation).
  • nCr is always equal to nC(n−r) — choosing r items is equivalent to choosing which (n−r) items to leave out.
  • nC0 and nCn are always 1, since there is exactly one way to choose nothing or everything.

Common Mistakes

  • Using combination when the problem actually requires ordering (that's a permutation).
  • Entering r greater than n, which describes an impossible selection.
  • Forgetting that swapping items within a chosen group does not create a new combination.

FAQs

How is combination different from permutation?

Combination counts selections where order doesn't matter (like choosing a committee), while permutation counts arrangements where order does matter (like assigning positions).

What does nCr mean?

nCr reads as "n choose r" — the number of ways to select r items from a total of n distinct items without regard to order.

What is nCr when r = 0 or r = n?

nC0 always equals 1 (there is exactly one way to choose nothing), and nCn also equals 1 (there is exactly one way to choose everything).

Is nCr the same as nC(n-r)?

Yes, choosing r items to include is mathematically equivalent to choosing (n−r) items to exclude, so nCr always equals nC(n−r).

Where are combinations used in real life?

Combinations are used in lottery odds, card game probabilities, team/committee selection, and any scenario counting groups rather than sequences.

Results are exact for n up to 170, limited by JavaScript number precision.

Last updated: July 25, 2026