AMC 8 2023 Problem 8
AMC8 2023 年第 8 题
基本计数原理★★☆☆☆
Lola, Lolo, Tiya, and Tiyo participated in a ping pong tournament. Each player competed against each of the other three players exactly twice. Shown below are the win-loss records for the players. The numbers 1 and 0 represent a win or loss, respectively. For example, Lola won five matches and lost the fourth match. What was Tiyo’s win-loss record?
Lola，Lolo，Tiya，Tiyo参加一场乒乓球锦标赛．每名选手与其他三名选手中的每个人比赛恰好两次. 下面显示的是选手的胜负记录. 数字1和0分别代表赢或输. 例如Lola一共赢了五场比赛，第四场比赛输了. 问Tiyo的胜负记录是怎样的?
| Player | Result |
|---|---|
| Lola | 111011 |
| Lolo | 101010 |
| Tiya | 010100 |
| Tiyo | ?????? |
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
A
解析 · Solution
Solution 1
We can calculate the total number of wins ( 1 's) by seeing how many matches were players, which is 12 matches played. Then, we can calculate the # of wins already on the table, which is 5 + 3 + 2 = 10, so there are 12 - 10 = 2 wins left in the mystery player. Now, we will make the key observation that there is only 2 wins ( 1 's) per column as there are 2 winners and 2 losers in each round. Strategically looking through the columns counting the 1 's and putting our own 2 1's when the column isn't already full yields . -edits apex304, lpieleanu In total, there will be games because there are ways to choose a pair of people from the four players. And, each player will play each other player exactly twice. Each of these 12 games will have 1 winner and 1 loser, so there will be a total of 12 1's and 12 0's in the win-loss table. Therefore, Tiyo will have 12-10=2 1's and 12-8=4 0's in his record. Now, all we have to do is figure out the order of these 1 's and 0 's. In every round, there are two games; the players are split into two pairs and the people in those pairs play each other. Thus, every round should have 2 winners and 2 losers which means that every column of the win-loss table should have 2 1's and 2 0's. Looking at the filled-in table so far, we see that columns 4 and 6 need one more 1, so Tiyo must have 1 's in those columns and 0 's gone in the others. Therefore, our answer is
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