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(Algebraic verification)
Start with C(n-1,k-1) + C(n-1,k). Expand using the formula: (n-1)! / ((k-1)!(n-k)!) + (n-1)! / (k!(n-k-1)!).
Common denominator: k!(n-k)!. First term becomes k × (n-1)! / (k!(n-k)!). Second term becomes (n-k) × (n-1)! / (k!(n-k)!).
Factor out: (n-1)! / (k!(n-k)!) × (k + n - k) = (n-1)! / (k!(n-k)!) × n = n! / (k!(n-k)!) = C(n,k).
