Binomial Coefficient Calculator

What is a binomial coefficient?

The binomial coefficient C(n, k), also written \(\binom{n}{k}\) and read "n choose k", counts the number of ways to choose k items from a set of n distinct items without regard to order. For example C(5, 2) = 10, because there are exactly 10 ways to pick 2 cards from 5 distinct cards.

The formula

The standard definition uses factorials: \[\binom{n}{k}=\frac{n!}{k!\,(n-k)!}\] A key symmetry property follows immediately: \(\binom{n}{k}=\binom{n}{n-k}\). Choosing which k items to include is equivalent to choosing which n − k items to leave out.

Pascal's rule

Every interior entry in Pascal's triangle satisfies the recurrence \[\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}\] with boundary conditions \(\binom{n}{0}=\binom{n}{n}=1\). This makes it easy to build the triangle row by row without computing factorials.

The binomial theorem

Binomial coefficients appear naturally as the coefficients in the expansion of a binomial power: \[(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\] For example \((a+b)^3=a^3+3a^2b+3ab^2+b^3\); the coefficients 1, 3, 3, 1 are exactly row 3 of Pascal's triangle.