256 teams play in a state soccer tournament. A team is elimi

Question

256 teams play in a state soccer tournament. A team is eliminated from the tournament after one loss. In the first round, all 256 teams play one game. If a team wins, it advances to the next round, where it plays another winning team. This process repeats itself until only one team is left, having advanced through each round without losing. How many games are played in the tournament?

A. 255

B. 256

C. \(\frac{256!}{(254!)(2!)(2!)}\)

D. \(\frac{256!}{(254!)(2!)}\)

E. \(\frac{256!}{254!}\)

Show Answer

Official Answer

A

A. 255

B. 256

C.  \frac{256!}{(254!)(2!)(2!)}

D.  \frac{256!}{(254!)(2!)}

E.  \frac{256!}{254!}

smyarga · Accepted Answer

In order to one team to win, all other teams should lose. So we can calculate the number of games as the number of team-losers. There are 255 such teams. The answer should be A.

P.S. If course you can calculate as sum of 128 games in first round, 64 in second, 32 in third, 16 in fourth, 8 in fifth, 4 in sixth, 2 in seventh, and 1 in eights. Still you get the same answer A. But not so fast.

smyarga · Answer

With pleasure. Every game has exactly one loser. To calculate the number of games is the same as calculate the number of losers. Every team except the winner loses only one game. So the number of games is the number of teams except the winner. Hope this helps.

retailingvnsupernova · Answer

Hi my friends, this is my solution to this problem
in round 1, there's  2^8  teams with  2^7  games are played.
and then in round 2, there's  2^7  teams left wiht  2^6  games are played
in the last round, there's  2  teams left with 1 game are played
so the games that is played is  A=2^7+2^6+...+2+1 
we must find the value of A
we have  2.A=2^8+2^7+...^2^2+2  and then  2.A-A=(2^8+2^7+...+2^2+2)-(2^7+2^6+...+2+1)=2^8-1 
so we have  A=2^8-1=255 
The answer is A
My friends, I think if you want to solve this problem fast, you should know that  256=2^8

smyarga · Answer

The most beautiful thing in this problem that you don't need to do any calculations. The answer will be always the number if teams-1.

retailingvnsupernova · Answer

Yes, after solving this problem as I posted above, I think that if you have  2^n  teams, there's will always be  2^n-1  games played
smyarga, please explain to me more clearly why there's 255 losers so we will have 255 games played. I think your solution is much faster and more clever than mine.

retailingvnsupernova · Answer

oh my god, my brain is too lazy :oops:

Thank you ^^

karishmatandon · Answer

i was over thinking this problem, thanks for explaining a quick and easy method..

Sergiy · Answer

We have simple set of geometrical progression with first element 128 (128 pairs = 256).
We have 8 rounds to final game, so A number of games (formula of sum for n elements of geometrical progression ) = (128* (q^8-1))/0,5-1 =
(128*(1-0,0625)(1+0,0625))/0,5 = First Impression that answer is 256, but you should look for slightly different answer such as 255
Where q- step of geometrical progression = 0,5!

RaunaqT · Answer

I followed the inputs for 8 teams and concluded that for 8 teams it will take 7 games similarly for 256 teams it will take 255 games

sahilvijay · Answer

lets say 3 teams 
A B and C
AB match => A wins
AC match => C wins

overall winner is C => Matches played AB and AC => 3-1 => n-1 if n teams 
So => 256 teams

255 is the answer.

gurmukh · Answer

Each match results in one team out and we have to send 255 teams out to have a winner so 255 matches are played. 
Hence option A is the answer

Posted from my mobile device

anks2190 · Answer

Please clarify the solution @e-gmat

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256 teams play in a state soccer tournament. A team is elimi

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