Combination & permutation calculator
Exact combinations C(n,k) and permutations P(n,r) with BigInt results, a log-based approximation for huge inputs, Pascal’s triangle, binomial expansion coefficients, combinations with repetition and multinomial coefficients.
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12034.787492720220301 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1
a^10 + 10a^9b + 45a^8b^2 + 120a^7b^3 + 210a^6b^4 + 252a^5b^5 + 210a^4b^6 + 120a^3b^7 + 45a^2b^8 + 10ab^9 + b^10
What this tool does
- Work out lottery or raffle odds: even large coefficients such as C(52,5) come back as an exact integer instead of a floating point approximation.
- Do statistics homework faster — C(n,k), P(n,k), combinations with repetition and multinomial coefficients in one place.
- Check a binomial expansion: the coefficients of (a + b)^n are listed term by term, matching Pascal’s triangle.
- Estimate enormous coefficients: once n reaches the tens of thousands the exact integer is useless, but the logarithm and the digit count still tell you the magnitude.
Example
Input
n = 10, k = 3, groups 2,2,1
Output
C(10, 3) = 120, P(10, 3) = 720, with repetition C(12, 3) = 220, ln C(10, 3) ≈ 4.787492, digits 3, multinomial 5!/(2!·2!·1!) = 30
Pascal’s triangle is drawn with n rows (at most 20) and the binomial expansion is shown for n ≤ 20.
Frequently asked questions
What is the difference between C(n, k) and P(n, k)?
Combinations ignore the order, permutations keep it. P(n, k) = C(n, k) × k!, so choosing 3 people out of 10 gives C(10,3)=120, while lining them up gives P(10,3)=720.
Is the result still accurate for large n?
Up to n = 2000 you get a digit-exact BigInt you can copy. Beyond that the coefficient has thousands of digits, so the tool reports ln C(n,k) and the number of decimal digits instead — those figures stay accurate, but no full integer is produced.
What happens when k is larger than n?
Following the usual combinatorial convention the result is 0 rather than an error, because you cannot choose k items out of n. The logarithmic form shows −∞.
When do I need combinations with repetition or multinomial coefficients?
Combinations with repetition C(n+k−1, k) apply when picking is allowed to repeat, for example 2 scoops from 3 flavours (6 ways). Multinomial coefficients count the ways to split a set into labelled groups, e.g. 30 ways to split 5 people into groups of 2, 2 and 1.
Why does Pascal’s triangle match the combinations?
Because C(n, k) = C(n−1, k−1) + C(n−1, k) is exactly the triangle rule: every entry is the sum of the two above it. The unit tests compare both algorithms entry by entry.
Keywords：combinationpermutationnCrnPrbinomial coefficientpascal trianglemultinomial组合数排列数杨辉三角二项式展开多项式系数可重复组合