1988 AMC 12 第 14 题

Question

对任意实数

a

和正整数

k

，定义

\binom ak= \frac{\begin{gathered} a(a-1)(a-2)\cdots\\ {}\cdot(a-(k-1)) \end{gathered}} {\begin{gathered} k(k-1)(k-2)\cdots\\ {}\cdot(2)(1) \end{gathered}}\text{。}

求

\binom{-\frac12}{100}\div\binom{\frac12}{100}\ \text{？}

For any real number

a

and positive integer

k,

define

\binom ak= \frac{\begin{gathered} a(a-1)(a-2)\cdots\\ {}\cdot(a-(k-1)) \end{gathered}} {\begin{gathered} k(k-1)(k-2)\cdots\\ {}\cdot(2)(1) \end{gathered}}.

What is

\binom{-\frac12}{100}\div\binom{\frac12}{100}\ ?

-199

-197

-1

197

199

约去相同的分母

100!

Cancel the identical denominator

100!

将剩余因子写成连续奇数，并寻找连锁消去

Write the remaining factors as consecutive odd integers and look for telescoping cancellation

约去

100!

后，分子各因子的绝对值为

1,3,\ldots,199

，而分母各因子的绝对值为

1,1,3,\ldots,197

。所有公共奇数因子相消，只剩绝对值

199

。分子有

100

个负因子，分母有

99

个，所以商为

-199

。

所以正确答案是 A。

After canceling

100!,

the numerator factors have magnitudes

1,3,\ldots,199,

while the denominator factors have magnitudes

1,1,3,\ldots,197.

All common odd factors cancel, leaving magnitude

199.

The numerator has

100

negative factors and the denominator has

99,

so the quotient is

-199.

Thus the correct answer is A.

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