AMC 8 2015 Problem 15
AMC8 2015 年第 15 题
基本计数原理★★★☆☆
At Euler Middle School, 198 students voted on two issues in a school referendum with the following results: 149 voted in favor of the first issue and 119 voted in favor of the second issue. If there were exactly 29 students who voted against both issues, how many students voted in favor of both issues?
在 Euler 中学，198 名学生参与了一次针对两项议题的全校投票，结果如下：149 名学生赞成第一项议题，119 名学生赞成第二项议题。如果恰好有 29 名学生对这两项议题都反对，那么有多少名学生对这两项议题都赞成？
- A.49
- B.70
- C.79
- D.99
- E.149
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
Let: be the number of students who voted in favor of the first issue, be the number of students who voted in favor of the second issue, be the number of students who voted in favor of both issues. We are given: A = 149, B = 119. 29 students voted against both issues, so the number of students who voted for at least one issue is: 198 - 29 = 169. By the principle of inclusion and exclusion: . Substitute known values: , , .
There are 198-149=49 students who voted against the first issue. Out of these, 49-29=20 voted for the second issue. We also find that 50 students voted against the second issue but in favor of the first in a similar way. This means that 149-50=119-20= students voted in favor of both issues.
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