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Evaluate the value of the alternating binomial sum 20C₀ - 20C₁ + 20C₂ - 20C₃ + ⋯ - 20C₁₀.

1. 0
2. 20C₁₀
3. -20C₁₀
4. (1)/(2) 20C₁₀

Correct answer: (1)/(2) 20C₁₀

Solution

Since sum_{k=0}^{20} (-1)^k C(20,k) = 0 and the row is symmetric, summing k=0 to 10 (including the middle term once) gives exactly half of C(20,10), i.e. (1/2) 20C10.

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