What this tool does

The calculator evaluates the binomial coefficient C(n,k): the number of ways to choose k items from n items when order does not matter and items are not repeated.

The calculation runs entirely in the browser and uses exact integer arithmetic.

How to use the tool

Enter the total number of available items as n.
Enter how many items should be selected as k.
The exact result appears automatically; if k is greater than n, the result is 0.
Copy the result or clear the fields before starting another calculation.

Useful applications

Counting unordered selections without repetition.
Solving combinatorics and probability exercises.
Finding coefficients used in the binomial theorem.
Checking mathematical or programming examples.

Notes and limitations

Only non-negative whole numbers from 0 to 10,000 are accepted.
The output can contain thousands of digits; BigInt keeps the integer exact.
C(n,k) counts combinations, so changing the order does not create a new selection.
For ordered selections, use a permutation formula instead.

Frequently asked questions

What does “n choose k” mean?

It is the number of ways to select k items from n available items without repetition and without considering order.

Why is the result 0 when k is greater than n?

You cannot select more items than are available, so there are no valid selections.

Why are C(n,k) and C(n,n-k) equal?

Choosing the items that are included is equivalent to choosing the items that are left out.

Can the calculator handle very large results?

Yes. It uses BigInt and returns the exact integer for inputs up to 10,000.

What is the difference between combinations and permutations?

Combinations ignore order; permutations treat different orders as different outcomes.

Reference

Formula, example and properties

The factorial formula removes the different orders of the same selection and therefore counts each combination once.

Binomial coefficient formula

C(n,k) = n! / (k! · (n-k)!)

n: Total number of available items.
k: Number of selected items.

The standard condition is 0 ≤ k ≤ n. Outside this range there are no valid selections.

The result is a count of unordered selections, not a list of the selections.

Complete example: C(5,2) = 10

C(5,2) = 5! / (2! · (5-2)!)
5! / (2! · 3!) = 120 / (2 · 6) = 10
There are 10 unordered ways to choose two items from five.

Symmetry: C(n,k) = C(n,n-k).
Boundary value: C(n,0) = 1.
Boundary value: C(n,n) = 1.

Combinations versus permutations

Use the binomial coefficient when order does not matter. If A-B and B-A count as different results, the problem requires permutations.