Calculate binomial coefficients or find missing n, k, or C(n,k) from two known values using the combination formula and factorials step by step.
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Binomial Coefficient Formula
The binomial coefficient counts how many ways you can choose k items from n total items when order does not matter. It is usually written as C(n, k), nCk, or n choose k.
- C(n, k) = the binomial coefficient
- n = the total number of items
- k = the number of items chosen
- ! = factorial, so n! means n × (n - 1) × ... × 2 × 1
Select Binomial coefficient and enter n and k. The calculator evaluates the formula with exact integer arithmetic, avoiding factorial overflow. The supported range is 0 ≤ k ≤ n ≤ 3,000.
Select Total items (n) to search integer values from k through 3,000. With k = 0 and coefficient 1, every nonnegative n works; the calculator identifies this nonunique case.
Select Items chosen (k) to find all integer solutions from 0 through n. Both symmetric solutions, k and n − k, are shown when distinct.
Common Binomial Coefficient Values
These values can help you check small calculations quickly.
| n | C(n, 0) | C(n, 1) | C(n, 2) | C(n, 3) | C(n, 4) |
|---|---|---|---|---|---|
| 4 | 1 | 4 | 6 | 4 | 1 |
| 5 | 1 | 5 | 10 | 10 | 5 |
| 6 | 1 | 6 | 15 | 20 | 15 |
| 7 | 1 | 7 | 21 | 35 | 35 |
| 8 | 1 | 8 | 28 | 56 | 70 |
Input Rules and Result Meaning
| Rule or property | What it means |
|---|---|
| n must be a nonnegative integer | You cannot choose from a negative number of items. |
| k must be a nonnegative integer | The number chosen must be 0 or greater. |
| k cannot be greater than n | You cannot choose more items than the total available. |
| C(n, k) = C(n, n - k) | Choosing k items is equivalent to leaving out n - k items. |
Examples
Example 1: Find the binomial coefficient
Find C(8, 3).
The binomial coefficient is 56.
Example 2: Find the missing value of n
Suppose k = 2 and the binomial coefficient is 45.
Testing integer values of n, you get:
So the missing value is n = 10.
FAQ
What does a binomial coefficient represent?
A binomial coefficient represents the number of combinations of k items that can be selected from n total items. Order does not matter. For example, choosing A then B is the same selection as choosing B then A.
Why can two different k values give the same binomial coefficient?
Because C(n, k) = C(n, n - k). For example, C(10, 3) = 120 and C(10, 7) = 120. Choosing 3 items to include is the same as choosing 7 items to leave out.
Why might there be no solution for a missing n or k?
There may be no integer value that matches the coefficient you entered. Binomial coefficients only occur at specific integer combinations of n and k, so not every number can be matched for a given input.
