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Math formula

How to calculate combination

A combination counts selections when order does not matter.

See the formulaOpen calculator
Direct answer
Formula
C(n, r) = n! / (r! * (n - r)!)

Use the table below to match each symbol with the right input. Keep units consistent before you start.

Combination formula
Start here when you only need the equation and variable names.
Formula
C(n, r) = n! / (r! * (n - r)!)

A combination counts selections when order does not matter.

Variable meanings
Check each symbol, meaning, and unit before you calculate.
SymbolNameDescriptionUnit
nTotal itemsNumber of items available.Depends on inputs
rItems chosenNumber of items selected.Depends on inputs
!FactorialProduct of whole numbers down to 1.Depends on inputs
When to use this formula
Check that your situation matches the formula before you trust the result.
  • Useful for committees, lottery picks, card hands, and groups where the same items in a different order count once.
Step-by-step method
Follow these steps when you are solving it by hand.
  1. 1Identify n, the number of available items.
  2. 2Identify r, the number selected.
  3. 3Calculate n!, r!, and (n - r)!.
  4. 4Divide n! by r! times (n - r)!.
Examples
These sample numbers show the order of operations and units.
Choose 3 from 10
n
10
r
3
  1. 1.C(10, 3) = 10! / (3! * 7!)
  2. 2.C(10, 3) = 720 / 6
  3. 3.C(10, 3) = 120
Result
There are 120 combinations.
Choose 2 from 6
n
6
r
2
  1. 1.C(6, 2) = 6! / (2! * 4!)
  2. 2.C(6, 2) = 30 / 2
  3. 3.C(6, 2) = 15
Result
There are 15 combinations.
Mistakes to avoid
Small input or unit errors can change the answer a lot.
  • Putting the wrong value in the numerator or denominator of the combination formula.
  • Skipping signs, order, or sorting rules that change the result.
  • Rounding before the last step.
  • Assuming a formula applies when the needed input is zero or missing.
Open the Probability Calculator
Open the calculator for the answer without rewriting the formula.

Enter your values in the related calculator, then compare the output with the hand method above.

Open Probability Calculator
FAQs
Short answers for common formula questions.
Can I calculate Combination by hand?

Yes. Write the units next to each value, convert rates or measurements first, and round only the final answer.

Why does my Combination result differ from another calculator?

Most differences come from rounding, unit conversions, rate timing, or a slightly different version of the formula.

When is the Probability Calculator better than hand math?

Open the Probability Calculator to check several scenarios or skip the hand arithmetic.