Math
Binomial Coefficient Calculator
The binomial coefficient C(n, k), read 'n choose k', counts the ways to pick k items from n and is the coefficient of x^k in (1 + x)^n.
Enter n and k and the calculator gives the exact value even when it has hundreds of digits, shows the shortened product after cancelling factorials, checks it with Pascal's identity C(n, k) = C(n - 1, k - 1) + C(n - 1, k), and prints the matching row of Pascal's triangle.
Whole number from 0 to 5000.
Whole number. If k is bigger than n the coefficient is 0.
Try an example
Result
C(n, k)
—
- Factorial formula
- —
- After cancelling
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- Pascal's identity
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- Symmetry
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- Row n of Pascal's triangle
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- Sum of row n = 2^n
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More details (1 more)
- Decimal
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Study path
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What the binomial coefficient calculator solves
C(n, k), also written (n k) in a tall bracket or nCk, is the number of k-element subsets of an n-element set. Order does not matter, which is why the k! in the The bottom number of a fraction; it says how many equal parts make one whole. removes the orderings. The same numbers appear as the The number multiplying a variable, like the 3 in 3x. in the binomial expansion (a + b)^n and as the entries of Pascal's triangle.
Example: C(10, 4)
- 10! / (4! 6!): cancel the 6! to leave (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1).
- 5040 / 24 = 210.
- Pascal check: C(9, 3) + C(9, 4) = 84 + 126 = 210.
- Row 10 of Pascal's triangle: 1 10 45 120 210 252 210 120 45 10 1; entry k = 4 is 210.
| n | Row of Pascal's triangle |
|---|---|
| 0 | 1 |
| 1 | 1 1 |
| 2 | 1 2 1 |
| 3 | 1 3 3 1 |
| 4 | 1 4 6 4 1 |
| 5 | 1 5 10 10 5 1 |
How to use it
- Enter n, the number of items, and k, the number chosen.
- Read C(n, k) and the cancelled product that produces it.
- Use the Pascal row to see the neighbouring coefficients, and the identity line to verify the value.
How to read the answer
C(n, k) is a whole number: the count of k-subsets, or the coefficient of a^(n-k) b^k in (a + b)^n. C(n, 0) = C(n, n) = 1, C(n, 1) = n, and the values grow toward the middle of the row. If k is outside 0 to n the coefficient is 0.
Common mistakes and edge cases
- Computing n! in full before cancelling, which overflows a calculator for n above about 20.
- Confusing combinations C(n, k) with permutations P(n, k) = n!/(n - k)!, which counts ordered selections.
- Forgetting that C(n, k) = C(n, n - k): choosing 4 of 10 equals choosing 6 of 10.
- Using k larger than n and expecting a positive answer; it is 0.
Worked examples
Typical
C(10, 4) = 210
C(n, k)
210
Poker hands
C(52, 5) = 2,598,960
C(n, k)
2,598,960
Edge k = 0
C(7, 0) = 1
C(n, k)
1
k = n
C(7, 7) = 1
C(n, k)
1
k larger than n
C(5, 8) = 0
C(n, k)
0
Large values stay exact
C(100, 50) = 100,891,344,545,564,193,334,812,497,256
C(n, k)
100,891,344,545,564,193,334,812,497,256
Negative n
n must be 0 or more
C(n, k)
Error
Frequently asked questions
What does n choose k mean?+
The number of ways to pick k items out of n when the order of picking does not matter. C(10, 4) = 210 means there are 210 different 4-item groups in a set of 10.
How is the binomial coefficient related to Pascal's triangle?+
Row n of the triangle lists C(n, 0), C(n, 1), …, C(n, n). Each entry is the sum of the two above it, which is Pascal's identity C(n, k) = C(n - 1, k - 1) + C(n - 1, k).
Why is C(n, k) equal to C(n, n - k)?+
Choosing which k items to take is the same as choosing which n - k items to leave behind. The formula shows it too: swapping k and n - k leaves k!(n - k)! unchanged.
How do I compute C(n, k) without huge factorials?+
Cancel (n - k)! from n!, leaving k factors on top: n(n - 1)…(n - k + 1), then divide by k!. Use the smaller of k and n - k to keep the product short.
What is C(n, k) when k is negative or bigger than n?+
0. There is no way to choose more items than exist, or a negative number of items.
Where do binomial coefficients appear in algebra?+
In the binomial theorem: (a + b)^n = Σ C(n, k) a^(n-k) b^k. For example (x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1 uses row 4 of Pascal's triangle.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
References
- OpenStax, College Algebra 2e, Section 9.6 “Binomial Theorem” — Binomial coefficients, Pascal's triangle, and the binomial theorem.
- OpenStax, College Algebra 2e, Section 9.5 “Counting Principles”
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Last updated: September 4, 2026