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Find the coefficient of x¹⁰ in the expansion of the sum S = sum_(r=0)²⁰ (-1)^r * C(20,r) * (x+1)^r * (1/2 + x)^(20-r).
- 0
- 1
- 2
- 3
Correct answer: 0
Solution
S = sum_(r=0)²⁰ C(20,r)*[-(x+1)]^r*(x+1/2)^(20-r) = (-(x+1) + (x+1/2))²⁰ = (-1/2)²⁰ = 1/2²⁰. This is a constant, so the coefficient of x¹⁰ is 0.
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