GMAT Club Olympics 2024 (Day 9): There are n teams in Group A

Question

There are n teams in Group A, and each team in Group A played each of the other teams exactly n times. What was the total number of games played by the teams in Group A?

(1) The number of games played by each team in Group A is more than 10 but fewer than 25.
(2) The number of games played by each team in Group A is half the total number of games played by all the teams in Group A.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Bunuel · Accepted Answer

GMAT Club Official Explanation:

There are n teams in Group A, and each team in Group A played each of the other teams exactly n times. What was the total number of games played by the teams in Group A?

There are  C^2_n  pairs possible from n teams, and since each pair played n times, the total number of games played is:

C^2_n*n=

=\frac{n!}{2!(n-2)!}*n=

=\frac{(n-2)!(n-1)n}{2(n-2)!}*n=

=\frac{(n-1)n}{2}*n=

=\frac{(n-1)n^2}{2} .

(1) The number of games played by each team in Group A is more than 10 but fewer than 25.

Each team played (n - 1) other teams n times, hence each team played (n - 1)n games. Hence, this statement implies that  10 < (n-1)n < 25 . By trial and error, we can find that two values of n satisfy this: 4 and 5. Not sufficient.

(2) The number of games played by each team in Group A is half the total number of games played by all the teams in Group A.

This statement implies that:

2*(n - 1)n=\frac{(n-1)n^2}{2}

2=\frac{n}{2}

n=4

Sufficient.

Answer: B.

divi22 · Answer

From statement 1:
10< n(n-1)< 25
which on solving gives n=4,5
Hence, it is not sufficient to answer the question.

From statement 2:
n(n-1) = n*n*(n-1)/4
which on solving gives n = 0,1,4
From which n=0,1 is illogical as n=0 means there is no team, and if n=1, then how can the matches be possible and it is illogical to put only one team in a group.

Hence n=4, which means it is SUFFICIENT to answer the question

The answer is B.

meet459 · Answer

There are n teams in Group A, and each team in Group A played each of the other teams exactly n times. What was the total number of games played by the teams in Group A?

No of games played by each team once with other team \(=n-1\)

No of games played by each team n times with each other team \(=n*(n-1)\)

Total no of games played when played once with each other team \(=\frac{n*(n-1)}{2}\)

Total no of games played when played n times with each other team \(=\frac{n*(n-1)}{2}*n\)

Statment 1: The number of games played by each team in Group A is more than 10 but fewer than 25.

\(10<n*(n-1)<25\)

when \(n=4\), \(n*(n-1)=12\)

when \(n=5\), \(n*(n-1)=20\)

when \(n=6\), \(n*(n-1)=30\)

\(n\) could be 4 or 5

NOT SUFFICIENT

Statement 2: The number of games played by each team in Group A is half the total number of games played by all the teams in Group A

\(n*(n-1)=\frac{n*(n-1)}{2}*\frac{n}{2}\)

\(4*n*(n-1)=n^2*(n-1)\)

\(n=4\)

SUFFICIENT

Answer is B

Kinshook · Answer

Given: There are n teams in Group A, and each team in Group A played each of the other teams exactly n times.
Asked: What was the total number of games played by the teams in Group A?

The number of games played by each team in Group A = n(n-1)
Total games played by all the teams in Group A = n * n(n-1)/2

(1) The number of games played by each team in Group A is more than 10 but fewer than 25.
10 < n(n-1) < 25
10 < n^2 - n < 25
3 < n < 6
n = {4, 5}
Total games played by all the teams in group A = {24, 50}
NOT SUFFICIENT

(2) The number of games played by each team in Group A is half the total number of games played by all the teams in Group A.
n(n-1) = n*n(n-1)/4; n = 4
Total games played by all the teams in Group A = n^2(n-1)/2 = 24
SUFFICIENT

IMO B

AthulSasi · Answer

Total number of ways to choose 2 teams out of n   = \frac{n (n-1)}{2}

When each of these pairs plays n times, total number of games, G  = n * \frac{n (n-1)}{2} 
 G = \frac{n^2  (n-1)}{2}

Number of games played by each team (g)
each team plays (n-1) other teams n times, 
g = n (n-1)

Statement 1
 The number of games played by each team in Group A is more than 10 but fewer than 25.  
says that g is more than 10 but less than 25. 
By trial and error we see that this is possible for n=4 as well as n=5
when n=4, g = 12
when n=5, g = 20

hence not sufficient.

Statement 1
 The number of games played by each team in Group A is half the total number of games played by all the teams in Group A.  
says that G/g = 2

i.e  \frac{\frac{n^2  (n-1)}{2}}{n (n-1)} = 2

n = 4

hence sufficient.

Correct answer B

PK1 · Answer

N teams in Group A, and each team in Group A player with each of other teams exactly n times
Total number of games played by team in Group A??

Option 1: The number of games played by each team in Group A is more than 10 but fewer than 25.
if we got n team and they play with one another once , total number of games become nC2
if each team plays with others once, number of games each team plays = (n-1)
so 10< n(n-1) < 25
so n could be 4 and 5
we do not get unique value

Option 2:The number of games played by each team in Group A is half the total number of games played by all the teams in Group A.
lets say each team plays with one another once
so (n-1)=  nC2/2
(n-1) = n(n-1)/4
so n = 4

if they play twice 
2(n-1) = 2* nC2/2
2(n-1)= n(n-1)/2
n= 4

if they play thrice
3(n-1) = 3* nC2/2
(n-1) = n (n-1)/4
n= 4

ntimes * (n-1)  = ntime * nC2/2
n = 4

so we have 4 teams playing 4 times. and total number of games played = 4C2 * 4 = 24

B is the answer

xhym · Answer

Each team plays n-1 other teams and they play each other team n times. That means that each team plays n(n-1) games. Statement 1 10 < n(n-1) < 25 Let's assume values of n, starting with 4: 10 < 4*3 < 25 -> 10 < 12 < 25 -> valid Now if n = 5: 10 < 5*4 < 25 -> 10 < 20 < 25 -> valid This means we can have multiple answers, so this is not sufficient. -> Statement 1 alone is not sufficient. Statement 2 n(n-1) games per team and n teams total so: Total games = \frac{n*n(n-1)}{2} (note that we must divide by 2 because each game is counted twice otherwise) We are told in the statement that n(n-1) is half the number of total games so: n(n-1) = \frac{1}{2} * \frac{n*n(n-1)}{2} 4*n(n-1) = n*n(n-1) 4 = n That means that n is 4 and if we plug it back into \frac{n*n(n-1)}{2} , we get total games played = 24. -> Statement 2 alone is sufficient. The answer is B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

prashantthawrani · Answer

There are n teams in Group A, and each team in Group A played each of the other teams exactly n times. What was the total number of games played by the teams in Group A?

Let teams in Group A be denoted as 
team  a        n    games with b,c....n]
team  b        n-1 games with c,d....n]
team  c        n-2 games with d,e,....n]
.
.
.
team  n-1      1 game with n]
team  n         0 game]

Total games in Group A= (n) + (n-1) + (n-2) + .....3 + 2 + 1

Total games in Group A=n(n+1)/2............(i)

(1) The number of games played by each team in Group A is more than 10 but fewer than 25.
10<n<25.
 Insufficient  to solve (i) to find Total games in Group A

(2) The number of games played by each team in Group A is half the total number of games played by all the teams in Group A.

n= /2 
n=n(n+1)/4
4n=n^2+n
n^2-3n=0
n(n-3)=0
 n=3  or n=0 (Impossible as Number of matches cant be zero)

Sufficient to solve (i) to find Total games in Group A

kavyavishnoi02 · Answer

Given that there are n teams in Group A and each team in Group A played each of the other teams exactly n times.

We need to find the total number of games played by all the teams in Group A.

Each team plays n games against each of the other n−1 teams. So,  each team plays n × (n−1) games.

Since each game involves two teams, if each team plays n × (n−1) games, then the  total number of games , considering both teams for each game, would be:   / 2​

Let's analyse each statement now:

(1)  10<n×(n−1)<25

Let's find possible integer values of n that satisfy this inequality:

For n=1: 1×(1−1)=0
For n=2: 2×(2−1)=2
For n=3: 3×(3−1)=6
For n=4: 4×(4−1)=12 (satisfies 10<12<25)
For n=5: 5×(5−1)=20 (satisfies 10<20<25)
For n=6: 6×(6−1)=30 (does not satisfy 10<30<25)

From this, n could be 4 or 5. However, this range does not give us a unique value for n.

Therefore,  Statement (1) alone is not sufficient .

(2)  Total number of games for all teams:  / 2​

According to Statement (2): n × (n−1) = 1/2 ×  /2​

n × (n−1) =  /4​

n=4

Thus, Statement (2) gives us a unique value for n.

Therefore,  Statement (2) alone is sufficient .

Hence, the answer is  option (B)

DG1989 · Answer

There are n teams in Group A, and each team in Group A played each of the other teams exactly n times. What was the total number of games played by the teams in Group A?

(1) The number of games played by each team in Group A is more than 10 but fewer than 25.
(2) The number of games played by each team in Group A is half the total number of games played by all the teams in Group A.

Solution:  Given that,
 
 There are 'n' teams in Group A 
 Each team plays each of the other teams exactly 'n' times  
The total number of games played by each team against every other team = n(n - 1)
Since each game involves two teams, the total number of games, say X played by all teams can be calculated as:
X =  \frac{n}{2}  * n(n -1)

X =  \frac{n^2(n - 1)}{2}

We need to determine the value of X

Statement 1:    The number of games played by each team in Group A is more than 10 but fewer than 25 
This means, 10 < n(n -1) < 25
if n = 3
n(n - 1) = 6 which is not > 10

if n = 4
n(n - 1) = 12 which lies between 10 and 25

if n = 5
n(n - 1) = 20 which lies between 10 and 25
Thus we are not getting a unique value of 'n' and cannot calculate X.
  INSUFFICIENT

Statement 2:    The number of games played by each team in Group A is half the total number of games played by all the teams in Group A 
This means, n(n -1) =  \frac{1}{2}  *  \frac{n^2(n - 1)}{2} 
On solving, we get n = 4
Thus, we can determine the value of X
  SUFFICIENT

The correct answer is Option B

OmerKor · Answer

Okay, we're almost at the end.
I will miss this competition, for sure.
The orange team looks invincible but we will fight until the end.
Let's start with our explanation of this topic:

Glance - the Question:  
 Question:  We are dealing with a combination and permutation problem.
We can solve this question by the formulate or by understanding.
I think it is better to understand it.

Rephrase - Reading and Understanding the question:  
 Given: 
Group A
It has n teams.
Each team play against each other n times. 
Let's understand it: We can take for example:
n=3, so 3 teams. played 3 games with each other. 
Team X play with Team Y  - 3 games
Team X play with Team Z  - 3 games
Team Y play with Team Z  - 3 games
So we have total of 9 games.
Each team played 6 games
With that in mind:

Plan  
Let continue with the examples:
n=4, so 4 teams. plays 4 games with each other. 
Team X play with Team Y  - 4 games
Team X play with Team Z  - 4 games
Team X play with Team T  - 4 games
Team Y play with Team Z  - 4 games
Team Y play with Team T  - 4 games
Team Z play with Team T  - 4 games
so total of 4x6 = 24 times
Each team played 12 times

n=5, Total of 50 times.
Each team played 20 times

Solve:  
(1)  10 < the number played by each < 25
so n can be 4 or 5 
n=4 each team played 12 times   
n=5 each team played 20 times
There   Insufficient   - We have to possible answers. n=4 or n=5

(2) Half of the total games played by each team
We can infer that n=4. Why?
When n=4.  the total games is 24 and each team played 12 games.
When n=5.  the total games is 50 and each team played 20 games. it's not half of 50.
Also we tried n=3 and we can see the pattern here, it increases each time, after 4 we cross the half. 
Thus,   Sufficient

Our Answer is B

THE END  
I hope you liked the explanation, I have tried my best here.
Let me know if you have any questions about this question or my explanation.
 https://cdn.gmatclub.com/cdn/files/forum/images/smilies/1f91e.png

Lizaza · Answer

Games played by each team  = N , games played total  = N^2 .

(1)
 N>10  and  N<25 , which can be pretty much any integer there. Eliminate.

(2)
 N = \frac{N^2 }{ 2} 
So  2N = N^2  and  N=2 
Sufficient.

Therefore,  the answer is B.

Suboopc · Answer

Games played by each team =  n*(n-1) 
Total number of games played by the teams =  \frac{n^2*(n-1)}{2}

Statement 1:
two possible values 
n = 4, 5 which lead to 12 and 20 games by each team

Statement 2:
 \frac{n^2*(n-1)}{4} = n*(n-1)  
3 possible values of n = 0, 1, 4
n cannot be 0 as there will be no teams to play.
n cannot be 1 since there will be no teams to play 1 game against.

Only possible value n = 4
Sufficient.

B

tgsankar10 · Answer

There are n teams in Group A, and each team in Group A played each of the other teams exactly n times. What was the total number of games played by the teams in Group A?

No of games played by each team once with other team  =n-1

No of games played by each team n times with each other team  =n*(n-1)

Total no of games played when played once with each other team  =\frac{n*(n-1)}{2}

Total no of games played when played n times with each other team  =\frac{n*(n-1)}{2}*n

Statment 1:    The number of games played by each team in Group A is more than 10 but fewer than 25.

10<n*(n-1)<25

when  n=4 ,  n*(n-1)=12

when  n=5 ,  n*(n-1)=20

when  n=6 ,  n*(n-1)=30

n   could be 4 or 5

NOT SUFFICIENT

Statement 2:    The number of games played by each team in Group A is half the total number of games played by all the teams in Group A

n*(n-1)=\frac{n*(n-1)}{2}*\frac{n}{2}

4*n*(n-1)=n^2*(n-1)

n=4

SUFFICIENT

Answer is B

bumpbot · Answer

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.

GMAT Club Olympics 2024 (Day 9): There are n teams in Group A

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

GMAT Club Olympics 2024 (Day 9): There are n teams in Group A

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards