About

The binomial coefficient C(n, k) counts the number of ways to choose k elements from a set of n without regard to order. It appears in probability distributions, polynomial expansions, and error-correcting codes. A miscalculated coefficient propagates through every downstream formula that depends on it. This tool computes exact results using native arbitrary-precision arithmetic, eliminating the floating-point truncation that plagues spreadsheet implementations for n > 66. It assumes n and k are non-negative integers with k ≤ n.

The multiplicative formula is used instead of the factorial definition to avoid intermediate overflow. The tool exploits the symmetry property C(n, k) = C(n, n − k) to minimize the number of multiplications. Pro tip: for very large n, verify your result against Kummer's theorem for carry counting in base-p arithmetic when you need the coefficient modulo a prime.

Formulas

The binomial coefficient is defined as the ratio of factorials. The factorial definition is:

C(n, k) = n!k! ⋅ (n − k)!

This tool uses the numerically stable multiplicative form to avoid computing large intermediate factorials:

C(n, k) = k∏i = 1 n − k + ii

Key identities used internally:

C(n, k) = C(n, n − k) (Symmetry)

C(n, k) = C(n − 1, k − 1) + C(n − 1, k) (Pascal's Rule)

n∑k = 0 C(n, k) = 2ⁿ (Row Sum)

Where n = total number of elements in the set, k = number of elements chosen, i = iteration index in the product, and C(n, k) = the number of k-combinations from n elements.

Reference Data

n	C(n,0)	C(n,1)	C(n,2)	C(n,3)	C(n,4)	C(n,5)	C(n,6)	C(n,7)	C(n,8)	C(n,9)	C(n,10)
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
6	1	6	15	20	15	6	1
7	1	7	21	35	35	21	7	1
8	1	8	28	56	70	56	28	8	1
9	1	9	36	84	126	126	84	36	9	1
10	1	10	45	120	210	252	210	120	45	10	1
12	1	12	66	220	495	792	924	792	495	220	66
15	1	15	105	455	1365	3003	5005	6435	6435	5005	3003
20	1	20	190	1140	4845	15504	38760	77520	125970	167960	184756
25	1	25	300	2300	12650	53130	177100	480700	1081575	2042975	3268760
30	1	30	435	4060	27405	142506	593775	2035800	5852925	14307150	30045015
40	1	40	780	9880	91390	658008	3838380	18643560	76904685	273438880	847660528
50	1	50	1225	19600	230300	2118760	15890700	99884400	536878650	2505433700	10272278170
100	1	100	4950	161700	3921225	75287520	1192052400	16007560800	186087894300	1902231808400	17310309456440

Frequently Asked Questions

Standard IEEE 754 double-precision floats lose integer precision beyond 2⁵³ (9007199254740992). For example, C(67, 33) equals 14226520737620288370, which already exceeds this limit. BigInt provides exact arbitrary-precision integer arithmetic, so results remain correct for n up to 10000 and beyond.

The multiplicative product runs in O(min(k, n − k)) time with O(1) extra space. The naive recursive Pascal's Rule has exponential time complexity O(C(n, k)) due to overlapping subproblems. Even memoized dynamic programming requires O(n ⋅ k) time and space, which is impractical for large inputs.

If k > n ÷ 2, the calculator replaces k with n − k before computing. This halves the maximum number of loop iterations. For instance, C(1000, 998) reduces to C(1000, 2), computing in 2 iterations instead of 998.

The generalized binomial coefficient extends to real and negative n using the Gamma function: C(n, k) = Γ(n + 1) ÷ (Γ(k + 1) ⋅ Γ(n − k + 1)). This tool restricts inputs to non-negative integers because exact BigInt arithmetic is only meaningful for the combinatorial (counting) interpretation.

The binomial theorem states (a + b)ⁿ = n∑k=0 C(n, k) a^n−k b^k. Each coefficient in the expansion is exactly the binomial coefficient for that term. Setting a = b = 1 yields the row-sum identity 2ⁿ.

C(0, 0) = 1 by convention: there is exactly one way to choose zero elements from an empty set. Similarly, C(n, n) = 1 and C(n, 0) = 1. If k > n, the result is 0 because you cannot choose more elements than available. All these boundary conditions are handled explicitly before the loop executes.