AMC 8 2012 Problem 14
AMC8 2012 年第 14 题
排列与组合★★★☆☆
In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference?
在名为“BIG N”的中学足球联盟中，每个队需要与其他队比赛一次。如果 2012 赛季总共进行了 21 场比赛，那么这个足球联盟有多少支球队？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
This problem is very similar to a handshake problem. We use the formula to usually find the number of games played (or handshakes). Now we have to use the formula in reverse. So we have the equation . Solving, we find that the number of teams in the BIG N conference is 7.
(If someone understands what I'm trying to do here and can explain it better, please edit it)We know that every team has to play a game with every other team, so we just need to find out how many consecutive numbers, 1 to , can fit into 21. We know that 6+5+4+3+2+1=21, and since this doesn't count to 7th team that shook hands with the other 6, we know that there are 7 teams in the BIG N conference. EDIT: So basically reiterating this, we are using a factorial and adding one because the team that they are talking about is not shaking the hands. Also as the other person said, we need to see how many consecutive numbers can add up to 21.
Lets say there are teams in total and we number the teams with letters. Team A plays x - 1 games against the other teams. Team B plays x - 1 minus the game it played with team A, making it x - 2 games. Team C plays x - 3 games, and so on. Finding the sum of the consecutive numbers 1 to x - 1 which equals 21 gives x = 7. So the correct answer choice is 7. A great way to visualize this problem is to draw circles and connecting lines between them. If you have 6 circles, you can draw 5 lines connecting to the first circle, then 4 more distinct lines connecting to the second circle, 3 more to the third, and so on until you have the second last circle drawing 1 line to the last circle.
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