pass@k for Code Generation

How good is a code model if you let it try k times? Generate n ≥ k samples per problem, count the c that pass the tests, and use the unbiased estimator from the Codex paper:

\text{pass@}k = 1 - \frac{\binom{n-c}{k}}{\binom{n}{k}}

Binomials overflow quickly for large n, so compute it as a product: $1 - \prod_{i=n-c+1}^{n} \left(1 - \frac{k}{i}\right)$ . If $n - c < k$ , the answer is exactly 1.0.

Implement pass_at_k(n, c, k) and mean_pass_at_k(results, k), where results is a list of (n, c) pairs, one per problem.