Home › Statistics & Probability Calculators › Combination Calculator (nCr) with Steps
Combination Calculator (nCr)
A combination is a selection where order does not matter. This calculator returns C(n, r) = n! ÷ [r!(n − r)!], the number of ways to choose r items from n distinct items, as an exact whole number. With repetition allowed, it counts multisets using C(n + r − 1, r).
Choosing 3 people from 10 for a committee gives C(10, 3) = 120: the 720 ordered line-ups divided by the 3! = 6 orders of each group. Lottery odds work the same way: a 6-from-49 draw has C(49, 6) = 13,983,816 possible tickets. Combinations with repetition answer questions such as how many ways there are to pick 6 donuts from 4 flavours, C(9, 6) = 84.
Combinations are the counting step inside the binomial and hypergeometric probabilities. Large values are computed exactly with integer arithmetic rather than rounded floating point, so C(300, 120) is shown with every digit.
Result and step-by-step solution
Ways to choose 3 from 10 when order does not matter.
- FormulaC(n, r) = n! ÷ [r! (n − r)!]
- SubstituteC(10, 3) = 10! ÷ (3! × 7!) = 120
- Check against permutationsP(10, 3) = 720, and each group of 3 can be ordered 3! ways, so C = P ÷ 3!
Frequently asked questions
- What is the formula for nCr?
- C(n, r) = n! ÷ [r! × (n − r)!]. For example C(10, 3) = 3,628,800 ÷ (6 × 5,040) = 120.
- Why is C(n, r) equal to C(n, n − r)?
- Choosing r items to take is the same as choosing n − r items to leave behind.
- How do I calculate combinations with repetition?
- Use C(n + r − 1, r). Picking 6 donuts from 4 flavours gives C(9, 6) = 84.
- How do I find nCr on a calculator?
- On a TI-84, enter n, press MATH, go to PRB, choose nCr and enter r. On a Casio use the nCr key (SHIFT + ÷ on many models).
Last reviewed: October 7, 2026. Calculations run in your browser and were validated against SciPy.