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Find the coefficient of x²⁰²⁰ in the expansion of (1 + x + x² + x³)¹⁰¹⁰ * (1 - x)¹⁰¹¹.
- 0
- 1010C505
- -1010C505
- None of these
Correct answer: -1010C505
Solution
Write (1+x+x²+x³)¹⁰¹⁰*(1-x)¹⁰¹¹ = (1-x)*[(1+x+x²+x³)(1-x)]¹⁰¹⁰ = (1-x)*(1-x⁴)¹⁰¹⁰. Since 2020=4*505, the coefficient of x²⁰²⁰ in (1-x⁴)¹⁰¹⁰ is C(1010,505)*(-1)⁵⁰⁵ = -C(1010,505), and the coefficient of x²⁰¹⁹ in (1-x⁴)¹⁰¹⁰ is 0 (2019 not divisible by 4). Therefore the coefficient of x²⁰²⁰ is -C(1010,505).
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