Question

12．設(shè)（1+x）³+（1+x）⁴+…+（1+x）⁵⁰=a₀+a₁x+a₂x²+…+a₅₀x⁵⁰，則a₃等于（　�。�

• A． C${\;}_{51}^{3}$
• B． C${\;}_{51}^{4}$
• C． 2C${\;}_{50}^{3}$
• D． C${\;}_{50}^{4}$

Answer 1

分析 （1+x）³中，含x³的系數(shù)為${C}_{3}^{3}$，（1+x）⁴中，含x³的系數(shù)為${C}_{4}^{3}$，…，（1+x）⁵⁰中，含x³的系數(shù)為${C}_{50}^{3}$，利用組合數(shù)的性質(zhì)${C}_{n}^{m}$+${C}_{n}^{m-1}$=${C}_{n+1}^{m}$即可得到答案．

解答 解：依題意，a₃=${C}_{3}^{3}$+${C}_{4}^{3}$+${C}_{5}^{3}$+…+${C}_{50}^{3}$
=（${C}_{4}^{4}$+${C}_{4}^{3}$）+${C}_{5}^{3}$+…+${C}_{50}^{3}$
=（${C}_{5}^{4}$+${C}_{5}^{3}$）+${C}_{6}^{3}$+…+${C}_{50}^{3}$
=${C}_{50}^{4}$+${C}_{50}^{3}$
=${C}_{51}^{4}$．
故選：B．

點(diǎn)評 本題考查二項式定理的應(yīng)用，著重考查組合數(shù)的性質(zhì)${C}_{n}^{m}$+${C}_{n}^{m-1}$=${C}_{n+1}^{m}$的應(yīng)用，屬于中檔題．