🎲 Estimating Binomial Coefficients with Entropy: A Beautiful Shortcut

Computing binomial coefficients like C(n, k) — the number of ways to choose k items from n — gets messy when n is large.

But there’s a surprisingly elegant shortcut from information theory:

C(n, k) ≈ 2ⁿʰ(p)
where p = k / n, and h(p) is the binary entropy function:

h(p) = -p · log₂(p) - (1 - p) · log₂(1 - p)