What Is Combination Notation?
Combination notation is shorthand for the number of ways to choose k items from n when order does not matter. You will see it written as C(n, k), nCk, nCr, ⁿCₖ or as the binomial coefficient with n stacked over k.
Type the combination in whichever notation you have. The calculator reads n and k, computes n! ÷ (k!(n − k)!), shows the symmetry C(n, n − k), the matching permutation count, and the row of Pascal's triangle for small n.
All of these notations mean the same number. Which one you use depends on your textbook, calculator or software.
How to Use the Combination Notation Calculator
- Type your combination in any common notation.
- Read the value and the equivalent forms.
- Use the steps and the Pascal row to see where it sits.
Combination Notation Formula
Combination formula
Symmetry
Relation to permutations
Combination Notation Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| n | Total number of items | |
| k or r | Number of items chosen | k ≤ n |
| n! | n factorial, the product 1 × 2 × … × n | |
| C(n, k) | Number of combinations | Also written nCk |
Combination Notation Worked Example
With the example values loaded in the calculator above, the steps are:
- Read the notation as n = 10, k = 3.
- C(n, k) = n! ÷ (k! × (n − k)!) = 10! ÷ (3! × 7!).
- Cancel the common factorial: (10 × 9 × 8) ÷ 3! = 120.
Result: C(10, 3) = 120.
Common ways to write a combination
| Notation | Read as | Where you see it |
|---|---|---|
| C(n, k) | n choose k | Most textbooks |
| nCk or nCr | n choose k | Calculators, school maths |
| ⁿCₖ | n choose k | Older textbooks |
| (n over k) as a stacked pair | binomial coefficient | Algebra and the binomial theorem |
Worked example: C(10, 3)
C(10, 3) = 10! ÷ (3! × 7!) = (10 × 9 × 8) ÷ (3 × 2 × 1) = 720 ÷ 6 = 120.
By symmetry C(10, 7) is also 120, since choosing 3 to include is the same as choosing 7 to leave out.
Where Combination Notation Is Used
Reading textbook notation
Convert nCr, C(n,r) and binomial coefficient notation into a number.
Counting problems
Find how many committees, hands or samples are possible.
Binomial theorem work
Get the coefficients for expanding (a + b)ⁿ.
Common Combination Notation Mistakes to Avoid
- If k is more than half of n, use the symmetry to make the arithmetic smaller.
- C(n, 0) and C(n, n) are always 1.
Combination Notation Terms and Definitions
- Binomial coefficient
- Another name for C(n, k), because it is the coefficient in the expansion of (a + b)ⁿ.
Combination Notation FAQs
What does C(n, k) mean?
It is the number of ways to choose k items from n without regard to order.
Are nCr, C(n,r) and n choose r the same?
Yes. They are three notations for the same value.
How do I calculate nCr by hand?
Compute n! ÷ (r! × (n − r)!), or cancel the common factorial first to keep the numbers small.
What is C(n, 0)?
Always 1, because there is exactly one way to choose nothing.
How is a combination different from a permutation?
A permutation counts ordered selections and equals a combination times k!.
What notations can I type?
C(10,3), 10C3, 10 choose 3, (10,3) and 10,3 all work.