Combinations Calculator (nCr)
Use this tool to calculate how many ways you can choose r items from n items when order does not matter.
What does “combinations” mean?
In probability and counting problems, a combination tells you how many ways you can select a group of items from a larger set when order does not matter. If you pick Alice, Ben, and Cara for a team, that is the same group as Cara, Alice, and Ben. Same people, same team, one combination.
This is different from permutations, where order matters. Combinations are used in statistics, lotteries, class committee selections, password analysis, and many everyday planning tasks.
The formula used by a combinations calculator
The standard notation is nCr, read as “n choose r.”
nCr = n! / (r! × (n − r)!)
Where:
- n = total number of items
- r = number of items chosen
- ! = factorial (for example, 5! = 5 × 4 × 3 × 2 × 1)
Example: If there are 10 books and you want to choose 3, then 10C3 = 120. That means there are 120 unique groups of 3 books.
How to do combinations on a scientific calculator
Typical key sequence
Most scientific calculators include an nCr function. The exact button location depends on the model, but usually:
- Enter n
- Press the nCr key (sometimes under a SHIFT/2nd function)
- Enter r
- Press equals
For instance, to calculate 12C4: type 12, press nCr, type 4, then press =.
If your calculator has no nCr key
You can still compute combinations manually with factorials:
- Find n!
- Find r!
- Find (n-r)!
- Divide n! by r!(n-r)!
That works, but it becomes tedious and error-prone for large values, which is why dedicated calculators and online tools are convenient.
Common mistakes when using combinations
- Mixing up permutations and combinations: If order matters, use nPr, not nCr.
- Using r > n: You cannot choose more items than the total available.
- Entering decimals: n and r should be whole numbers in standard counting problems.
- Forgetting that 0 is valid: nC0 is always 1.
Quick practice examples
Example 1: Class committee
A class has 20 students. How many ways can 4 students be chosen for a committee?
20C4 = 4,845 ways.
Example 2: Lottery selection
You choose 6 numbers from 49 in a lottery. The number of possible tickets is:
49C6 = 13,983,816.
Example 3: Choosing interview candidates
From 8 applicants, choose 2 for final interviews:
8C2 = 28 possible pairs.
When to use combinations in real life
- Building teams or committees
- Card game probability calculations
- Sampling methods in statistics
- Planning unique bundles of products
- Estimating odds in random draws
Final thoughts
Combinations are one of the most useful counting tools in mathematics and decision-making. Once you know that order does not matter, nCr is often the right method. Use the calculator above to get accurate answers fast, and always double-check that your inputs are whole numbers with r less than or equal to n.