Question

2．${C}_{n}^{o}$+${C}_{n+1}^{1}$+${C}_{n+2}^{2}$+…+${C}_{n+m-1}^{m-1}$=${C}_{n+m}^{m-1}$．

Answer 1

分析 利用组合数公式${C}_{n}^{m}$+${C}_{n}^{m-1}$=${C}_{n+1}^{m}$，进行化简即可．

解答 解：${C}_{n}^{o}$+${C}_{n+1}^{1}$+${C}_{n+2}^{2}$+…+${C}_{n+m-1}^{m-1}$=${C}_{n+1}^{0}$+${C}_{n+1}^{1}$+${C}_{n+2}^{2}$+…+${C}_{n+m-1}^{m-1}$
=${C}_{n+2}^{1}$+${C}_{n+2}^{2}$+…+${C}_{n+m-1}^{m-1}$
=${C}_{n+3}^{2}$+…+${C}_{n+m-1}^{m-1}$
=…=${C}_{n+m-1}^{m-2}$+${C}_{n+m-1}^{m-1}$
=${C}_{n+m}^{m-1}$．
故答案为：${C}_{n+m}^{m-1}$．

点评 本题考查了组合数公式的应用问题，也考查了逻辑思维与推理能力，是基础题目．