2009 AMC 12B 第 6 题

Question

通过插入括号，可以使表达式

2 \times 3 + 4 \times 5

取得若干个值。可以得到多少个不同的值？

By inserting parentheses, it is possible to give the expression

2 \times 3 + 4 \times 5

several values. How many different values can be obtained?

2

3

4

5

6

结果取决于加法相对于两个乘法在什么时候进行。

The result depends on when the addition is performed relative to the two multiplications

系统列出不同的分组方式；有些加括号方式会得到相同的值。

Enumerate the distinct groupings; several parenthesizations collapse to the same value

真正不同的分组给出

这些分组给出

(2 \times 3) + (4 \times 5) = 26

、

\ ((2 \times 3) + 4) \times 5 = 50

、

\ 2 \times (3 + (4 \times 5)) = 46

、

\ 2 \times (3 + 4) \times 5 = 70

。

这四个值互不相同，所以可以得到

4

个值。

所以正确答案是 C。

The genuinely different groupings give

(2 \times 3) + (4 \times 5) = 26,

\ ((2 \times 3) + 4) \times 5 = 50,

\ 2 \times (3 + (4 \times 5)) = 46,

and

\ 2 \times (3 + 4) \times 5 = 70.

These are all distinct, so

4

values can be obtained.

Thus, the correct answer is C.

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