AMC12 2009 B

AMC12 2009 B · Q6

AMC12 2009 B · Q6. It mainly tests Manipulating equations, Casework.

By inserting parentheses, it is possible to give the expression \[2\times3 + 4\times5\] several values. How many different values can be obtained?

通过插入括号，可以使表达式 \[2\times3 + 4\times5\] 具有多个值。可以得到多少个不同的值？

(A) 2 2

(B) 3 3

(D) 5 5

(E) 6 6

Answer

Correct choice: (C)

正确答案：（C）

Solution

The three operations can be performed on any of $3! = 6$ orders. However, if the addition is performed either first or last, then multiplying in either order produces the same result. So at most four distinct values can be obtained. It is easy to check that the values of the four expressions \begin{align*} (2\times3) + (4\times5) &= 26,\\ (2\times3 + 4)\times5 &= 50,\\ 2\times(3 + 4\times5) &= 46,\\ 2\times(3 + 4)\times5 &= 70 \end{align*} are in fact all distinct. So the answer is $\boxed{4}$, which is choice $\mathrm{(C)}$.

这三个运算可以按任意 $3! = 6$ 种顺序进行。然而，如果加法要么最先进行要么最后进行，那么两次乘法无论按哪种顺序进行都会得到相同结果。因此最多只能得到四个不同的值。容易检验下面四个表达式的值 \begin{align*} (2\times3) + (4\times5) &= 26,\\ (2\times3 + 4)\times5 &= 50,\\ 2\times(3 + 4\times5) &= 46,\\ 2\times(3 + 4)\times5 &= 70 \end{align*} 确实都互不相同。所以答案是 $\boxed{4}$，对应选项 $\mathrm{(C)}$。

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