Factorial Calculator
Compute n! with step-by-step multiplication, permutations, combinations, double factorial, Stirling approximation, trailing zeros, digit count, growth visualization, and a reference table up to 25!.
Binomial Coefficient Calculator
Calculate binomial coefficients C(n,k), Pascal's triangle rows, permutations P(n,k), and multinomial coefficients with step-by-step factorial breakdowns.
C(10, 3)⧉
120.00
Combinations: 10! / (3! × 7!) = 120.00
P(10, 3)⧉
720.00
Permutations: 10! / (7!) = 720.00
n!⧉
3,628,800.00
10! = 3,628,800.00
k!⧉
6.00
3! = 6.00
(n−k)!⧉
5,040.00
(10−3)! = 7! = 5,040.00
Multinomial Coefficient⧉
N/A
Enter k values that sum to n (10). Currently: 3 = 3
C(n,k) / P(n,k) Ratio⧉
0.166667
Ratio = 1/k! — shows how many permutations map to each combination
Sum of Pascal's Row⧉
1,024.00
2^10 = 1,024.00 — sum of all C(10, 0..10)
Pascal's Triangle Row 10 — Visual
1.00
k=0
10.00
k=1
45.00
k=2
120.00
k=3
210.00
k=4
252.00
k=5
210.00
k=6
120.00
k=7
45.00
k=8
10.00
k=9
1.00
k=10
Pascal's Triangle (first 5 rows)
| Row | Coefficients |
|---|---|
| 0 | 1.00 |
| 1 | 1.00 1.00 |
| 2 | 1.00 2.00 1.00 |
| 3 | 1.00 3.00 3.00 1.00 |
| 4 | 1.00 4.00 6.00 4.00 1.00 |
Factorial Breakdown
| Expression | Value |
|---|---|
| 10! | 3,628,800.00 |
| 3! | 6.00 |
| (10−3)! = 7! | 5,040.00 |
| C(10,3) | 120.00 |
| P(10,3) | 720.00 |
Planning notes, formulas, and examples
About the Binomial Coefficient Calculator
The binomial coefficient, written as C(n, k) or "n choose k," counts the number of ways to select k items from a set of n distinct items without regard to order. It appears throughout mathematics — in the Binomial Theorem for expanding (a + b)^n, in probability when calculating the chance of k successes in n trials, and in Pascal's triangle where each entry is a binomial coefficient.
The formula is C(n, k) = n! / (k! · (n − k)!). For example, C(5, 2) = 120 / (2 · 6) = 10, meaning there are exactly 10 ways to pick 2 items from 5. This calculator computes not only C(n, k) but also the related permutation count P(n, k) = n! / (n − k)!, the full Pascal's triangle row for n, and multinomial coefficients when you need to split items into more than two groups.
Understanding binomial coefficients is essential for students of algebra, probability, statistics, and computer science. Whether you're expanding polynomial expressions, counting lottery combinations, or analyzing algorithm complexity, the page gives you the result together with the factorial structure behind it.
When This Page Helps
Binomial coefficients are simple to state but easy to mis-handle once factorial growth becomes large or you need to compare combinations, permutations, and multinomial counts. This page is useful because it shows n!, k!, (n−k)!, the main coefficient, and the related counts together, so you can validate the structure of the calculation instead of trusting a single number.
How to Use the Inputs
- Enter n (total items) and k (items chosen) in the input fields.
- Select the mode, method, or precision options that match your binomial coefficient problem.
- Read n! first, then use k! to confirm your setup is correct.
- Try a preset such as "C(5,2)" to test a known case quickly.
Formula used
C(n, k) = n! / (k! × (n − k)!); P(n, k) = n! / (n − k)!; Multinomial = n! / (k₁! × k₂! × … × kₘ!)Example Calculation
Result: n! shown by the calculator
Using the preset "C(5,2)", the calculator evaluates the binomial coefficient setup, applies the selected algebra rules, and reports n! with supporting checks so you can verify each transformation.
Tips & Best Practices
- C(n, k) = C(n, n−k), so C(10,3) = C(10,7). Use whichever is easier.
- For large n, the calculator avoids overflow by cancelling common factors.
- Pascal's triangle row n contains all C(n, 0) through C(n, n).
- The sum of all entries in Pascal's triangle row n is 2^n.
How This Binomial Coefficient Calculator Works
This calculator takes n (total items), k (items chosen), k₂ (optional, for multinomial), k₃ (optional, for multinomial) and applies the relevant binomial coefficient relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Interpreting Results
Start with the primary output, then use n!, k!, (n−k)!, Multinomial Coefficient to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
Study Strategy
Sources & Methodology
Last updated:
Frequently Asked Questions
Combinations C(n,k) count selections where order does not matter. Permutations P(n,k) count arrangements where order does matter. P(n,k) = C(n,k) × k!.
Pascal's triangle is a triangular array where each entry is the sum of the two entries above it. Row n of the triangle contains the binomial coefficients C(n,0), C(n,1), …, C(n,n).
A multinomial coefficient extends the binomial to more than two groups. It equals n! / (k₁! × k₂! × … × kₘ!) and counts ways to split n items into groups of sizes k₁, k₂, …, kₘ.
In the standard combinatorial definition, both n and k must be non-negative integers with k ≤ n. Generalized binomial coefficients allow real or negative n, but this calculator uses the standard definition.
There is exactly one way to choose zero items from any set — by choosing nothing. Mathematically, 0! = 1 by definition, so C(n,0) = n! / (0! × n!) = 1.
JavaScript can safely represent integers up to 2^53 − 1. For very large n, the factorial values overflow, but the calculator handles moderate values (n up to ~170) accurately.
More in this topic
Permutation Calculator
Calculate permutations P(n, r) — the number of ordered arrangements of r items from n.
Probability Calculator
Calculate basic probability for single and combined events. Compute AND, OR, NOT probabilities and conditional probability.