lnCk

lnCk denotes the natural logarithm of the binomial coefficient C(n, k) = n choose k, the number of ways to select k elements from an n-element set, with 0 ≤ k ≤ n. The quantity is commonly used in probability and statistics to work with probabilities and likelihoods in log-space.

Mathematically, ln C(n, k) can be expressed in several equivalent ways. It equals ln(n!) − ln(k!) − ln((n − k)!). It can also be written using the gamma function as ln Γ(n+1) − ln Γ(k+1) − ln Γ(n−k+1). A numerically stable form for computation is ln C(n, k) = ∑_{i=1}^k [ln(n − k + i) − ln i].

For large n with p = k/n, Stirling-based approximations give ln C(n, k) ≈ n H(p) − (1/2) ln(2π n p(1 − p)) + o(1), where H(p) = −p ln p − (1 − p) ln(1 − p) is the natural entropy function. This connects the log-binomial coefficient to information-theoretic concepts and provides a compact asymptotic description.

Domain and usage notes: ln C(n, k) is defined for integers n ≥ 0 and 0 ≤ k ≤ n. If k is outside this range, C(n, k) is zero and ln C(n, k) is undefined. In practice, ln C(n, k) is used to compute log-probabilities in binomial models, to compare combinatorial counts without overflow, and to facilitate asymptotic analyses in combinatorics and statistics.

Example: C(5, 2) = 10, so ln C(5, 2) = ln(10) ≈ 2.302585.