2009 AMC 10B 第 7 题

Question

通过插入括号，表达式

2\times3+4\times5

可以得到若干个值。能得到多少个不同的值？

By inserting parentheses, it is possible to give the expression

2\times3+4\times5

several values. How many different values can be obtained?

2

3

4

5

6

判断哪一个运算最后进行。

Decide which operation is performed last

尝试诸如

(2\times3)+(4\times5)

和

(2\times3+4)\times5

的分组。

Try groupings such as

(2\times3)+(4\times5)

and

(2\times3+4)\times5

四个数和三个运算共有五种完整的加括号方式。逐一计算可得

((2\times3)+4)\times5=50

，而

(2\times(3+4))\times5=70

。

(2\times3)+(4\times5)=26

。

2\times((3+4)\times5)=70

，而

2\times(3+(4\times5))=46

。

因此不同的结果是

26, 46, 50

和

70

，共

4

个。

所以正确答案是 C。

There are five full parenthesizations of four numbers joined by three operations. Evaluating all five gives

((2\times3)+4)\times5=50

and

(2\times(3+4))\times5=70.

(2\times3)+(4\times5)=26.

2\times((3+4)\times5)=70

and

2\times(3+(4\times5))=46.

Thus the distinct values are

26, 46, 50,

and

70,

for a total of

4.

Thus, the correct answer is C.

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