Question

2．已知f（x）=（2x-1）¹⁰=a₁₀x¹⁰+a₉x⁹+a₈x⁸+…+a₁x+a₀，則C${\;}_{2}^{2}$a₂+C${\;}_{3}^{2}$a₃+C${\;}_{4}^{2}$a₄+…+C${\;}_{10}^{2}$a₁₀=180．

Answer 1

分析 根據(jù)f（x）的展開式，求出a₂、a₃、a₄、…、a₁₀的值，
再計算C${\;}_{2}^{2}$a₂+C${\;}_{3}^{2}$a₃+C${\;}_{4}^{2}$a₄+…+C${\;}_{10}^{2}$a₁₀的值．

解答 解：f（x）=（2x-1）¹⁰
=（1-2x）¹⁰
=1-2${C}_{10}^{1}$x+2²${C}_{10}^{2}$x²-…+（-1）^r•2^r•${C}_{10}^{r}$•x^r+…+2¹⁰•x¹⁰
=a₁₀x¹⁰+a₉x⁹+a₈x⁸+…+a₁x+a₀，
∴C${\;}_{2}^{2}$a₂+C${\;}_{3}^{2}$a₃+C${\;}_{4}^{2}$a₄+…+C${\;}_{10}^{2}$a₁₀
=${C}_{2}^{2}$•2²•${C}_{10}^{2}$-${C}_{3}^{2}$•2³•${C}_{10}^{3}$+${C}_{4}^{2}$•2⁴•${C}_{10}^{4}$-…+${C}_{10}^{2}$•2¹⁰•${C}_{10}^{10}$
=180-2880+20160-80640+201600-322560+322560-184320+46080
=180．

點評 本題考查了組合數(shù)的公式與二項式展開式的應用問題，也考查了計算能力，是易錯題．