There are exactly 6 teams in league x. What was the total

Question

There are exactly 6 teams in league x. What was the total number of games played by the 6 teams last season?

(1) Each team in league x played each of the other teams at least once.

(2) No team in league x played more than 7 games.

Bunuel · Accepted Answer

There are exactly 6 teams in league x. What was the total number of games played by the 6 teams last season?

(1) Each team in league x played each of the other teams at least once --> if each team played each of the other teams ONLY once then there was total of  C^2_6=15  games played (notice that each team played 5 games) but if each team played each of the other teams TWICE then there was total of  2*C^2_6=30  games played (notice that each team played 10 games). Not sufficient.

(2) No team in league x played more than 7 games. Clearly insufficient.

(1)+(2) If each team played each of the other teams ONLY once then there was total of  C^2_6=15  games played (each of the 6 teams played 5 games), but if after each team played each of the other teams ONLY once, some two teams played between each other once more then the total number of games played was 16 (4 teams played 5 games and 2 teams played 6 games). Not sufficient.

Answer: E.

NGGMAT · Answer

Hi Bunnel

I get confused in at least and at most qs... here st 1 says that each team played atleast one game with the other team. I understand that this mean some could have played more than one.
but in the other qs about out of 2 atleast 1 snake is black and 1 snake is white... we understand it as 1 black snake... 
please explain about qs with atleast atmost in the qs stem.

Bunuel · Answer

At least one means 1 or more; At most 1 means 1 or less. Nothing more in it. I think the question you are talking about is this one: devil-s-dozen-129312-20.html#p1063888 But this question is different and has some other constrains. P.S. If you have problems with most of the questions one of the reasons could be lack of fundamentals. So, I'd advice to go through the basics once more.

sri30kanth · Answer

Dear Bunuel,

Can you explain "but if after each team played each of the other teams ONLY once, some two teams played between each other once more then the total number of games played was 16" what this means? Thanks

Bunuel · Answer

Say each team played each of the other teams once: the number of games played =  C^2_6=15 . After that, say two of the teams played one more game between each other. The total number of games played = 15 + 1 = 16.

Hope it's clear.

sri30kanth · Answer

Yeah let's assume two of the teams played one more game between each other but still the total no of games a team played would be less than 7. Isn't it? Or can't we guarantee that? Say A,B,C,D,E,F are six teams. After initial 15 games where each team played 5 times, is there a possibility of other teams playing randomly? Please clarify. Thank you

Bunuel · Answer

Not sure I understand your question...

Each team played each of the other teams once, then two team played one more game between each other (say that was the final game). Total games = 16. Each team played less than 7 games. What's unclear here?

profileusername · Answer

Can you please help me understand the logic behind this combination application " C^2_6=15 " used when each team plays the other only once?

Thanks.

Bunuel · Answer

The total number of games possible will be total number of  different pairs  possible out of 6 teams (one game per one pair), so  C^2_6=15 .

Bunuel · Answer

ameyaprabhu · Answer

Hi, I am little confused about the use of the formula because according to the GMAT club Math book, the formula for combinations is  C^n_k=  n! / k! (n-k)!

Bunuel · Answer

6C2,  C^2_6 ,  C^6_2  all mean the same thing - choosing different groups of 2 out of 6:  \frac{6!}{2!4!} . Can it be anything else? Can we choose 6 out of 2?

stepanyan · Answer

Dear Bunuel, 
Could you pls clarify -- using what formula did you calculated number of played games? 
"(1) Each team in league x played each of the other teams at least once --> notice that each team played 5 games"

Bunuel · Answer

The total number of teams is six. If each team played every other team exactly once, then the total number of games would equate to the number of pairs that can be formed from these six teams. Mathematically, this is represented as 6C2, which results in 15 total games. When we look at the number of games each team played, it's five. This is because, out of the six teams, each has five potential opponents.

stepanyan · Answer

Bunuel, 
thanks a lot!

Komal324 · Answer

Dear  Bunuel

Hope you are well. Firstly, thank you for answering all my queries promptly. Really appreciate it. 
Secondly, I just wanted to tell you how I interpreted the question and got the right answer E. Kindly let me know if you find any gaps in my understanding.

Statement 1:    Each team in league x played each of the other teams at least once  
 This means for instance, team 1 can play with 6 other teams at most 6 times, but we do not know how many times. It could be Team 1 played with 3 teams or 4 teams, we do not know. In addition, in the question it is nowhere mentioned that each team played once with the other team so, that is why we will have multiple number of games played from one team. The same will be applied to other teams as well. Not sufficient because we will get multiple answers for the number of games played (n).

Statement 2:   No team in league x played more than 7 games.  
 It is the same situation as given in the first statement. If the number of games is n then n<7 then, it could be Team 1 played with 3 teams or 4 teams, we do not know. Hence, not sufficient.

Combine 1 and 2-  We are not getting any unique value of number of games played (n). Therefore, the answer is E.

Thank you!

Best, 
 Komal

Bunuel · Answer

In (1), it says  each team played  each  of the other teams at least once , so we actually know that every team played against all 5 other teams, meaning each team played at least 5 games. So it’s not that a team could have played with only 3 or 4 teams, that part is fixed. The uncertainty is whether teams played only once against each other or more than once.

There are exactly 6 teams in league x. What was the total

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There are exactly 6 teams in league x. What was the total

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