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There are 8 teams in a certain league and each team plays each of the other teams exactly once. If each game is played by 2 teams, what is the total number of games played?

A. 15 B. 16 C. 28 D. 56 E. 64

We are given that there are 8 teams in a league and that each game is played by 2 teams. Note that each team does not play itself, and the order of pairing each team with its opponent doesn't matter. [For example, the pairing of (Team A vs. Team B) is identical to the pairing of (Team B vs. Team A).] The situation can therefore be solved by finding the number of combinations of 8 items taken 2 at a time, or 8C2, as follows:

8C2 = 8! / [2! x (8-2)!]

(8 x 7 x 6!) / (2! x 6!)

(8 x 7)/2!

(8 x 7)/ 2

4 x 7 = 28

Answer C
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Re: There are 8 teams in a certain league and each team plays [#permalink]

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27 Jun 2017, 01:58

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Re: There are 8 teams in a certain league and each team plays [#permalink]

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27 Jun 2017, 06:43

Top Contributor

sarb wrote:

There are 8 teams in a certain league and each team plays each of the other teams exactly once. If each game is played by 2 teams, what is the total number of games played?

A. 15 B. 16 C. 28 D. 56 E. 64

There are 8 teams. If we ask each team, "How many teams did you play?" we'll find that each team played 7 teams, which gives us a total of 56 games (since 8 x 7 = 56).

From here we need to recognize that each game has been COUNTED TWICE. For example, if Team A and Team B play a game, then Team A counts it as a game, and Team B ALSO counts it as a game.

So, to account for the DUPLICATION, we'll divide 56 by 2 to get 28

Re: There are 8 teams in a certain league and each team plays [#permalink]

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27 Jun 2017, 13:49

There are 8 teams in a certain league and each team plays each of the other teams exactly once. If each game is played by 2 teams, what is the total number of games played?

Total number of games played can be well given by the combination formula - \(C^2_{8}\)

\(C^2_{8}\)

\(= \frac{8!}{2! * 6!}\)

\(= \frac{8 * 7 * 6!}{2 * 1 * 6!}\)

\(= 4 * 7\)

\(= 28\)

Hence, Answer is C _________________

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Re: There are 8 teams in a certain league and each team plays [#permalink]

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04 Jul 2017, 19:34

There is one approach which is the quickest one:

If we ask each team how many games it played, each team will say 7. Hence total 7×8=56 games are expected. However, each game has been counted twice, thus 56/2 = 28

The other way is to ask each team how many matches it has played such that each match is unique. The answer will be 7+6+5+4+3+2+1 = 28

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