The binomial coefficient C(n,k)
is defined as the number of different ways to choose a kelement subset from an nelement set.
The values C(n,k) appear in
Pascal's triangle and satisfy the recurrence
C(n,k) =
C(n − 1, k − 1) + C(n − 1, k).
Equivalently, C(n,k) is the coefficient of the a^{k}b^{n−k}
term in the full expansion of the binomial power (a + b)^{n}.
Note that in the expression (a + b)^{n} the variables a and b appear in a symmetric manner;
therefore, we have C(n,k) = C(n, n−k) for any
k ≤ n.
For example, the expansion
(a + b)^{4} =
a^{4} + 4a^{3}b + 6a^{2}b^{2} + 4ab^{3} + b^{4}
yields the following binomial coefficients:
C(4,0) = 1,
C(4,1) = 4,
C(4,2) = 6,
C(4,3) = 4,
C(4,4) = 1.
This online calculator computes binomial coefficients C(n,k)
for input values 0 ≤ k ≤ n ≤ 50000 in arbitrary precision arithmetic.
So, for instance, you will get all digits of C(9000,4500) –
all the 2708 digits of this very large number!
See also:
• 100+ digit calculator: arbitrary precision arithmetic
• Prime factorization calculator
• Euler's totient function φ calculator
• Highly composite numbers
• Divisors and sumofdivisors calculator
• Fibonacci numbers calculator
• Catalan numbers calculator

