Combination Calculator (nCr)
Enter two whole numbers to compute how many ways to choose r items from n items where order does not matter.
What is a combination?
A combination is a way of selecting items from a group where the order does not matter. For example, selecting 3 toppings for a pizza from 8 options is a combination problem, because choosing mushrooms, onions, and olives is the same set regardless of the order you list them.
The combination formula
The standard formula is:
C(n, r) = n! / (r!(n-r)!)
- n = total number of items available
- r = number of items selected
- ! = factorial, meaning multiplication down to 1
This calculator computes the exact whole-number result using integer arithmetic, so you can handle both small and fairly large inputs with precision.
How to use this calculator
- Enter a whole number for n.
- Enter a whole number for r.
- Click Calculate nCr.
Constraints: both values must be non-negative integers, and r ≤ n. For performance, this page currently allows values up to 5000.
Examples
Example 1: Choosing 2 from 5
C(5,2) = 10. There are 10 unique pairs you can form from 5 items.
Example 2: Lottery-style pick
C(49,6) = 13,983,816. That means there are over 13 million possible 6-number combinations from 49 numbers.
Example 3: Card hands
C(52,5) = 2,598,960. This is the number of unique 5-card poker hands from a standard 52-card deck.
Combination vs. permutation
People often mix up combinations and permutations:
- Combination: order does not matter (nCr)
- Permutation: order matters (nPr)
If you care about ranking, sequence, or arrangement, use permutation. If you only care about the chosen set, use combination.
Where combinations are used in real life
- Probability and statistics
- Card and board game analysis
- Lottery odds
- Sampling in surveys and experiments
- Feature selection in machine learning
Quick tips
- If r = 0 or r = n, then C(n,r) = 1.
- C(n,r) = C(n,n-r), so choosing 2 from 10 equals choosing 8 from 10.
- Always validate that r is not greater than n.