Question

2．組合數(shù)${C}_{34}^{0}$+${C}_{34}^{2}$+${C}_{34}^{4}$…+${C}_{34}^{34}$被9除的余數(shù)是8．

Answer 1

分析 根據(jù)組合數(shù)的特征，得出${C}_{34}^{0}$+${C}_{34}^{2}$+${C}_{34}^{4}$…+${C}_{34}^{34}$=2³³=（9-1）¹¹，
利用二項(xiàng)展開(kāi)式即可求出該組合數(shù)被9除的余數(shù)．

解答 解：∵${C}_{34}^{0}$+${C}_{34}^{2}$+${C}_{34}^{4}$…+${C}_{34}^{34}$=${C}_{34}^{1}$+${C}_{34}^{3}$+${C}_{34}^{5}$+…+${C}_{34}^{33}$，
∴${C}_{34}^{0}$+${C}_{34}^{2}$+${C}_{34}^{4}$…+${C}_{34}^{34}$=$\frac{1}{2}$×2³⁴
=2³³
=8¹¹
=（9-1）¹¹
=${C}_{11}^{0}$•9¹¹-${C}_{11}^{1}$•9¹⁰+${C}_{11}^{2}$•9²+…+（-1）^r•${C}_{11}^{r}$•9^r+…-${C}_{11}^{11}$•9⁰
=k×9-1
=（k-1）9+8，其中k∈N；
∴該組合數(shù)被9除的余數(shù)是8．
故答案為：8．

點(diǎn)評(píng) 本題考查了組合數(shù)公式的應(yīng)用問(wèn)題，也考查了二項(xiàng)式定理的應(yīng)用問(wèn)題，是基礎(chǔ)題目．