Question

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?

$6$

$7$

$8$

$9$

$10$

Answer: B

Concepts:combinations triangular number

Difficulty rating: 1100

Video solution:

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Written solution:

Each of the $N$ teams plays $N-1$ games. However, $2$ teams play each game, so multiplying $N$ and $N-1$ would be twice the number of games. Therefore, we know $N(N-1) = 42.$ This leads to $N^2 -N +0.25 = 42.25$ which implies $(N-0.5)^2 = 6.5^2.$ In turn, this suggests: $N-0.5 = 6.5$ $N = 7$

Thus, the answer is B .

2012 AMC 8 Problem 14

Answer: B

Difficulty rating: 1100

Video solution:

Written solution:

Problem 14 in Other Years