Question

1．（1+x）+（1+x）²+（1+x）³+（1+x）⁴+…+（1+x）⁴⁹展开式中x³的系数是（　　）

• A． ${C}_{51}^{3}$
• B． ${C}_{50}^{4}$
• C． ${C}_{51}^{4}$
• D． ${C}_{47}^{4}$

Answer 1

分析 根据题意得出（1+x）+（1+x）²+（1+x）³+（1+x）⁴+…+（1+x）⁴⁹展开式中x³的系数是${C}_{3}^{3}$+${C}_{4}^{3}$+${C}_{5}^{3}$+…+${C}_{49}^{3}$=${C}_{4}^{4}$+${C}_{4}^{3}$+${C}_{5}^{3}$+…+${C}_{49}^{3}$，利用组合数的性质求值即可．

解答 解：（1+x）+（1+x）²+（1+x）³+（1+x）⁴+…+（1+x）⁴⁹展开式中x³的系数是
${C}_{3}^{3}$+${C}_{4}^{3}$+${C}_{5}^{3}$+…+${C}_{49}^{3}$=${C}_{4}^{4}$+${C}_{4}^{3}$+${C}_{5}^{3}$+…+${C}_{49}^{3}$
=${C}_{5}^{4}$+${C}_{5}^{3}$+…+${C}_{49}^{3}$
=${C}_{6}^{4}$+…+${C}_{49}^{3}$
=..=${C}_{49}^{4}$+${C}_{49}^{3}$=${C}_{50}^{4}$．
故选：B．

点评 本题考查了二项式定理的应用问题，也考查了组合数的性质与应用问题，是基础题目．