AMC10 2018 B

AMC10 2018 B · Q3

AMC10 2018 B · Q3. It mainly tests Basic counting (rules of product/sum), Casework.

In the expression $( \times ) + ( \times )$ each blank is to be filled in with one of the digits 1, 2, 3, or 4, with each digit being used once. How many different values can be obtained?

在表达式$( \times ) + ( \times )$中，每个空白处填入数字1、2、3或4各一次。能得到多少不同的值？

(A) 2 2

(B) 3 3

(D) 6 6

(E) 24 24

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Both the multiplications and the addition can be performed in either order, so each possible value can be obtained by putting the 1 in the first position and one of the other three numbers in the second position. Therefore the only possible values are $(1 \times 2) + (3 \times 4) = 14,$ $(1 \times 3) + (2 \times 4) = 11,$ and $(1 \times 4) + (2 \times 3) = 10,$ so just 3 different values can be obtained.

答案（B）：乘法和加法都可以以任意顺序进行，因此每一个可能的值都可以通过把 1 放在第一个位置，并把另外三个数中的一个放在第二个位置来得到。因此唯一可能的值是 $(1 \times 2) + (3 \times 4) = 14,$ $(1 \times 3) + (2 \times 4) = 11,$ 以及 $(1 \times 4) + (2 \times 3) = 10,$ 所以只能得到 3 个不同的值。

Topics