Binomial Coefficient Calculator -- n Choose k

The binomial coefficient $C(n, k)$, also written as “n choose k,” counts the number of ways to choose $k$ items from $n$ items without regard to order. It equals $\frac{n!}{k! \times (n - k)!}$ and appears as the coefficients in the expansion of $(a + b)^n$.

The binomial theorem states that $(a + b)^n = \sum C(n, k) \times a^{n-k} \times b^k$, summed from $k = 0$ to $n$. Each term's coefficient is the corresponding binomial coefficient from row $n$ of Pascal's triangle.

The binomial distribution uses $C(n, k)$ to count outcomes. The probability of exactly $k$ successes in $n$ independent trials, each with probability $p$, is $C(n, k) \times p^k \times (1-p)^{n-k}$. For example, getting exactly 3 heads in 5 fair coin flips has $C(5,3) = 10$ favorable outcomes.

Instead of computing full factorials (which overflow for large $n$), use $C(n, k) = \prod \frac{n - k + i}{i}$ for $i$ from 1 to $k$. Each intermediate result is always an integer, making this method practical for large values.

Key identities include symmetry $C(n, k) = C(n, n-k)$, Pascal's recurrence $C(n, k) = C(n-1, k-1) + C(n-1, k)$, the row sum $\sum C(n, k) = 2^n$, and Vandermonde's identity $C(m+n, r) = \sum C(m, k) \times C(n, r-k)$.