Optimise nCk calculation using symmetry property

[orien]

Context

The nCk (binomial coefficient) calculation computes combinations using an iterative approach that loops from 1 to k. When k is large (e.g., calculating C(100, 95)), this results in many unnecessary iterations. The binomial coefficient has a mathematical symmetry property: C(n,k) = C(n,n-k), meaning the same result can be obtained by using the smaller value.

The symmetry exists because:

C(n,k) = n! / (k! × (n-k)!) = n! / ((n-k)! × k!) = C(n, n-k)

Changes

Modified the nCk method in lib/zxcvbn/math.rb to use the smaller of k or (n-k):

# Use symmetry property: C(n,k) = C(n, n-k)
# Choose smaller k to minimize iterations
k = n - k if k > n - k

This reduces loop iterations for large k values:

• C(100, 95) now calculated as C(100, 5) - 5 iterations instead of 95
• C(52, 47) now calculated as C(52, 5) - 5 iterations instead of 47
• C(20, 15) now calculated as C(20, 5) - 5 iterations instead of 15

Consequences

Performance improvement: 6.6x faster (1.694µs → 0.256µs per calculation) for test cases with large k values.

All 291 tests pass, including existing tests that verify the symmetry property (C(10,3) = C(10,7) and C(20,5) = C(20,15)). The optimization uses a proven mathematical property and maintains identical behaviour whilst significantly improving performance.