AMC 8 2005 Problem 14
AMC8 2005 年第 14 题
基本计数原理★★★☆☆
The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled?
小十二篮球联赛有两个赛区，每个赛区有六支球队。每支球队在各自的赛区内与其他球队比赛两次，且和另一个赛区内的每支球队比赛一次。则一共安排了多少场比赛？
- A.80
- B.96
- C.100
- D.108
- E.192
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
Within each division, there are pairings, and each of these games happens twice. The same goes for the other division so that there are 4(15)=60 games within their own divisions. The number of games between the two divisions is (6)(6)=36. Together there are 60+36=96 conference games.
Each team plays 10 games in its own division and 6 games against teams in the other division. So each of the 12 teams plays 16 conference games. Because each game involves two teams, there are games scheduled.
Suppose division 1 has teams A, B, C, D, E, F. Suppose division 2 has teams G, H, I, J, K, L. Then, for interdivisional games in Division 1, has 10 games scheduled. Because has already played twice in those 10 games, will not have to account for , making its scheduled games 8. This continues until has 0 games scheduled. The same will happen for division 2 in interdivisional games. So, 2(10 + 8 + 6 + 4 + 2) = 60. Now, for cross-divisional games, team will play $G, H, I, J, K,$ and once. Each team from division 1 plays 6 cross-divisional games, for a total of games. Division 2 will NOT be counted like done for division 1 since we have already considered the games. This is the same concept applied for interdivisional games. (This is also the concept applied in the Handshake Formula.) Now, 30 + 30 + 36 = 96 games scheduled.
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