Integer restrictions

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yellowho: Master | Next Rank: 500 Posts; Posts: 233; Joined: Wed Aug 22, 2007 3:51 pm; Location: New York; Thanked: 7 times; Followed by:2 members

Integer restrictions

by yellowho » Thu Jan 20, 2011 2:54 am

There are 12 teams in a round-robin soccer tournament, where each team plays every other team
once. The winning team of each game receives 3 points and the losing team receives 0 points. If
there is a tie, each team receives 1 point. If the total number of points given to all teams is 185,
how many games ended in a tie?

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Source: Beat The GMAT — Problem Solving | Thu Jan 20, 2011 2:54 am

jaxis: Master | Next Rank: 500 Posts; Posts: 101; Joined: Sun Sep 26, 2010 9:33 am; Thanked: 5 times

by jaxis » Thu Jan 20, 2011 3:11 am

12 teams soccer tournament,
each team plays every other team once in 12C2 ways = 66

now,
each winner gets 3 pts
Loser gets 0 pts
tie - each team gets one pt each.

If we have a winner then total pts awarded = 3+0 = 3
If we have a tie total pts awarded = 1+1 = 2

therefore , for each tie we get 1 pt decrease in total.
66 games should ideally have 66*3 = 198 pts awarded.
But due to ties..total pt awarded are 185.

==>no of ties = 198-185 = 13

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Thu Jan 20, 2011 3:12 am

yellowho wrote:There are 12 teams in a round-robin soccer tournament, where each team plays every other team once. The winning team of each game receives 3 points and the losing team receives 0 points. If there is a tie, each team receives 1 point. If the total number of points given to all teams is 185, how many games ended in a tie?

Total number of matches = Number of possible different selection of 2 teams out of 12 = 12C2 = 66

Say, x matches ended with a win or loss and y matches ended in tie. Hence, (x + y) = 66 .......... (1)

Now, for the x matches, winning team got 3 points and losing team 0 points. Hence per match a total of 3 points. But for the y tie matches, both team got 1 point each. Hence per match a total of 2 points.

Thus, (3x + 2y) = 185 ........ (2)

Multiplying (1) with 3 then subtracting (2) from it,