AMC8 2014

AMC8 2014 · Q16

AMC8 2014 · Q16. It mainly tests Basic counting (rules of product/sum), Combinations.

The "Middle School Eight" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?

“中学八强”篮球联赛有 $8$ 支队伍。每个赛季，每支队伍要与其他每支联赛队伍进行两场比赛（一场主场、一场客场），每支队伍还要与非联赛对手进行 $4$ 场比赛。涉及“中学八强”队伍的赛季总比赛场数是多少？

(A) 60 60

(B) 88 88

(D) 144 144

(E) 160 160

Answer

Correct choice: (B)

正确答案：（B）

Solution

Within the conference, there are 8 teams, so there are $\dbinom{8}{2}=28$ pairings of teams, and each pair must play two games, for a total of $28\cdot 2=56$ games within the conference. Each team also plays 4 games outside the conference, and there are 8 teams, so there are a total of $4\cdot 8 =32$ games outside the conference. Therefore, the total number of games is $56 + 32 = \boxed{\text{(B) }88}$.

联赛内有 $8$ 支队伍，因此共有 $\dbinom{8}{2}=28$ 对对阵组合，每对组合需要进行两场比赛，联赛内总比赛场数为 $28 \cdot 2 = 56$ 场。每支队伍还要与联赛外的对手进行 $4$ 场比赛，共有 $8$ 支队伍，因此联赛外总比赛场数为 $4 \cdot 8 = 32$ 场。因此，总比赛场数为 $56 + 32 = \boxed{\text{(B) }88}$ 场。

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