AMC 8 1985 Problem 9

AMC8 1985 年第 9 题

表达式与运算★★☆☆☆

The product of the 9 factors

9 个因子之积

A. $\displaystyle \frac {1}{10}$
B. $\displaystyle \frac {1}{9}$
C. $\displaystyle \frac {1}{2}$
D. $\displaystyle \frac {10}{11}$
E. $\displaystyle \frac {11}{2}$

正确答案 · Correct Answer

A

解析 · Solution

Solution 1

Simplifying each factor and cancelling yields

\displaystyle \Big(1 - \frac12\Big)\Big(1 - \frac13\Big)\Big(1 - \frac14\Big)\cdots\Big(1 - \frac {1}{10}\Big) =\left(\frac{1}{2}\right)\left(\frac{2}{3}\right)\left(\frac{3}{4}\right)\cdots\left(\frac{9}{10}\right) = \frac{1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8\cdot9}{2\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8\cdot9\cdot10} = \frac{1}{10}

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