Describe your change

• Fix a bug or typo in an existing algorithm?
• Add or change doctests?

Checklist

• I have read CONTRIBUTING.md.
• This pull request is all my own work -- I have not plagiarized.
• I know that pull requests will not be merged if they fail the automated tests.
• This PR only changes one algorithm file. To ease review, please open separate PRs for separate algorithms.
• All new Python files are placed inside an existing directory.
• All filenames are in all lowercase characters with no spaces or dashes.
• All functions and variable names follow Python naming conventions.
• All function parameters and return values are annotated with Python type hints.
• All functions have doctests that pass the automated testing.
• All new algorithms include at least one URL that points to Wikipedia or another similar explanation.

Summary

In maths/binomial_coefficient.py, binomial_coefficient(n, r) used:

if 0 in (n, r):
    return 1

Because of this check, binomial_coefficient(0, r) returned 1 for any r > 0 (such as binomial_coefficient(0, 10) == 1).

Mathematically, the binomial coefficient $\binom{n}{r}$ represents the number of ways to choose $r$ elements from a set of $n$ elements. When $r > n \ge 0$, $\binom{n}{r} = 0$. For all other cases where $r > n$ (e.g. binomial_coefficient(5, 6) or binomial_coefficient(3, 5)), the function correctly returned 0. Only the $n = 0, r > 0$ case returned 1 incorrectly due to 0 in (n, r).

Changes

1. Replace if 0 in (n, r): return 1 with standard boundary handling:

   if r == 0:
       return 1
   if r > n:
       return 0

   This ensures $\binom{0}{0} = 1$, $\binom{n}{0} = 1$, and $\binom{n}{r} = 0$ for all $r > n$ (including $n = 0$).
2. Update doctests to reflect $\binom{0}{0} = 1$ and $\binom{0}{10} = 0$, matching math.comb.

Testing

• Verified against math.comb(n, r) for all $n \in [0, 50]$ and $r \in [0, 50]$ (2,500 test cases with 0 mismatches).
• pytest maths/binomial_coefficient.py --doctest-modules passed.
• ruff check maths/binomial_coefficient.py passed.
• ruff format --check maths/binomial_coefficient.py passed.