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655-705 (Hard)|   Combinations|         
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GurpreetMahli
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 17 Jul 2026, 13:40
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Answer: Both Statements together are sufficient.

To solve this question we need to know how all 17 students are distributed in the clubs so we can pick teams of 3 that won't have any members that represent the same club.

Given:
Each of the 17 are in exactly one club and there is 3 or more clubs.

9 members are in chess or drama leaving 8 volunteer club affiliations unknown.

Statement One:
Tell us that that there are 3 members undistributed than can be distributed in the following way
3 2 3 or 3 2 2 1 or 3 2 1 1 1

Statement Two:
Tell us that of the 8 there is one club with less than 3
So there is either a club of 3 4 1 or 3 3 2

Statement One and Two
Both statement together leaves only one way to distribute the remaining 8 as 3 3 2 which would then allow us chose teams of 3 that don't have members in the same club because we know how every one is distributed among the clubs.
Bunuel
At a school festival, 17 students volunteered to lead activity booths. Each volunteer represented exactly one school club, and the volunteers represented at least 3 clubs. Among the volunteers, 5 represented the Drama Club and 4 represented the Chess Club. If a 3-student team is to be formed so that no two team members represent the same club, in how many ways can the team be formed?

(1) Among the 17 volunteers, 3 represented the Robotics Club and 2 represented the Art Club.
(2) Among the 17 volunteers, exactly one club was represented by fewer than 3 volunteers.


 


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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 17 Jul 2026, 14:28
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Bunuel
At a school festival, 17 students volunteered to lead activity booths. Each volunteer represented exactly one school club, and the volunteers represented at least 3 clubs. Among the volunteers, 5 represented the Drama Club and 4 represented the Chess Club. If a 3-student team is to be formed so that no two team members represent the same club, in how many ways can the team be formed?

(1) Among the 17 volunteers, 3 represented the Robotics Club and 2 represented the Art Club.
(2) Among the 17 volunteers, exactly one club was represented by fewer than 3 volunteers.


 


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Correct Answer C

Statement 1 : it leaves us with 3 (17 - 5 - 4 - 3 - 2) unaccounted volunteers, we do not know if they are in one club or split between clubs. Hence, giving us various combinations. Not Sufficient.

Statement 2 : it leaves us with 8 (17 - 5 - 4) unaccounted members, we are told one club as less than 2 members so the distribution could in any manner 6,2 or 3,3,2. Hence, giving us various combinations. Not sufficient.

Statement 1 + 2 : we know one club has less than 3 members, which is the arts club and now, the 3 unaccounted members from statement 1 cannot be in smaller groups of 1 or 2, they will have to be in a single group. Hence, we can calculate the exact combinations. Sufficient.
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 17 Jul 2026, 14:49
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Classic combination case. Now, in A you will get to know about 14 people only, 3 will have no terms. it can give lots of different answers. Statement 2nd also gives the same result; you will not find any unique value. Moving ahead, if we merge both, we will get unique values of in case of combinations as all population of clubs is known. C is the ans, both sufficient together.
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 17 Jul 2026, 17:31
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The question want to know the number of ways the teams can be formed consisting of 3 students so that no two team represent the same club.
St1: Of the 17 volunteers we know Robotics:3 Art:2 Drama:5 and Chess:4 Still we don't know what club does the remaining 3 member represent hence st1 is not sufficient
ST2: Exactly 1 cub by <3 volunteers: That also is not sufficient to answer
St1 and St2 together:
Case1: <3 students 1 club could be Art club only and we still dont know what happened with the last 3 students.
Case2: <3 students representing 1 club could be activity x, so it could be 1 student left or 2 students left out of 3 students with unknown club so in this case also we cant come to a clear conclusion.
Hence option E is correct
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 17 Jul 2026, 20:39
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I first got confused reading the question, the question asks that what are the total number of ways to make 3 (NEW) member-team from the total clubs available such that no two students are from the same club. If we could find the exact number of clubs and exact number of students in those clubs then I don't think we need to do any calculations because it would lead us to definitive answer.

Given:
Clubs -DramaChess atleast 1 other club shall be there
Number of
students		54


S1:
Club name -DramaChessRoboticsArtsAny number of clubs
number of students5432

So the students in each different clubs could be:
[5,4,3,2,2,1] , [5,4,3,2,3] - from each we could have different values of answer
NOT SUFFICIENT

S2:
it just says only one club with fewer than 3 members,
Cases possible: [5,4,6,2] , [5,4, 7,1]
NOT SUFFICIENT

S1 and S2 together
[5,4,3,2, remaining 3 members so we would have only one choice to form 1 club with 3 members]
Only possible case = [5,4,3,2,3]

Hence this would give us definitive answer.

IMO C
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 17 Jul 2026, 21:06
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Bunuel
At a school festival, 17 students volunteered to lead activity booths. Each volunteer represented exactly one school club, and the volunteers represented at least 3 clubs. Among the volunteers, 5 represented the Drama Club and 4 represented the Chess Club. If a 3-student team is to be formed so that no two team members represent the same club, in how many ways can the team be formed?

(1) Among the 17 volunteers, 3 represented the Robotics Club and 2 represented the Art Club.
(2) Among the 17 volunteers, exactly one club was represented by fewer than 3 volunteers.


 


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drama - 5, chess - 4
we don't know total number of clubs to make the team
from (1) 3 - robo, 2 - arts
3 students are left , there are multiple ways to divide them so not sufficient
from (1) + (2)
only one club was represented by less than 3 volunteers, so it is arts.
so 4 students have to be in same group
thus, clubs will be 5,4,3,2,3
Both are sufficient together.
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 17 Jul 2026, 21:38
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17 total volunteers, drama 5, chess 4, leaving 8 unaccounted. A team takes one student from each of three different clubs, so the answer is a sum of size products across every trio.

(1) Robotics 3, Art 2, so 14 placed and 3 floating. One club of 3 gives sizes 5, 4, 3, 3, 2. Three singleton clubs gives 5, 4, 3, 2, 1, 1, 1,. Different counts, insufficient.
(2) Exactly one club under 3. Drama and chess clear this so the small club lives in the 8 and the rest split into clubs of 3+. options include 7+1 and 6+2 which give different counts. insufficient.

Together Art's 2 is the one undersized club, so the remaining 3 can't split into anything smaller than 3. Sizes are 5, 4, 3, 3, 2. Sufficient
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 17 Jul 2026, 21:59
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Statement 1: Robotics = 3, Art = 2, Drama = 5, and Chess = 4. 3+2+5+4 = 14, so only 14 volunteers are accounted for leaving three who could be any any combination of clubs. Not sufficient

Statement 2: One club has fewer than 3 members doesn't tell us the number of club sizes to use for the probabilities.

Together. We know that D = 5, Chess =4, R = 3, and Art = 2. If only one club has fewer than three then any other club must be >=3. There are only 3 undefined volunteers so the last club must have three members so the group sizes are 5,4,3,3,2. Then you can use combinations to pick 3 students from 3 different clubs and see the number of ways. Both statements together are sufficient. C
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 17 Jul 2026, 22:08
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We know Drama = 5, Chess = 4 and Total Volunteers = 17.

We need the exact number of 3-person teams, with each student from a different club.

S1 --> Robotics = 3 and Art =2

This accounts for 14 volunteers, leaving 3 unassigned. Those 3 could form: one club of 3 or three clubs of 1 or two clubs (2 and 1). Each gives a different number of teams.

S1 - Not sufficient

S2 --> We are told exactly one club has fewer than 3 volunteers.

Multiple distributions are still possible, so the number of teams is not fixed

S2 - Not sufficient

Together, from S1 - Art has 2 volunteers, and from S2 - Art is the only club with fewer than 3 volunteers.

So, the remaining 3 volunteers must form one club of 3. The club sizes are fixed:
5, 4, 3, 3, 2

Now the number of teams is uniquely determined. So, S1 and S2 together are sufficient.

Answer: Option C
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 17 Jul 2026, 22:41
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Stem:
Total students = 17
Each student in only 1 club and there are at least 3 clubs
Drama club (D) = 5 and Chess club (C) = 4. Remaining undistributed students = 8
To know the ways in which teams can be formed, we need to know the number of clubs. That will give us total unique ways of selecting 3 clubs (since from each club, only 1 member can be picked, we need to first know the ways in which 3 clubs can be picked). Based on the club combinations, the ways of selecting 3 students can be derived.

St1:
Robotics club (R) = 3 and Art club (A) = 2
We still have 3 students remaining which can be divided in 3 clubs (1,1,1) or 2 clubs (2,1) or in 1 club alone (3). Hence NOT SUFFICIENT

St2:
1 club was represented by <3 students. So the students in this club can be 1 or 2. This will leave us with undistributed students = 7 or 6. These 7 or 6 students can then be divided into multiple combinations of clubs (3,3,1 or 2,2,2,1), which will keep on changing the number of clubs. Hence, NOT SUFFICIENT

Combined:
We know D=5, C=4, R=3 and A=2. This brings the total number of distributed students = 14. Then, the number of undistributed students = 3. Since statement 2 says only 1 club has <3 students, that club is Arts club. Hence the fifth club (F) can have only 3 students, which makes the total number of clubs = 5. Now there are 5C3 = 10 ways of picking 3 unique clubs and then depending on which clubs are picked, total ways of picking students can be uniquely identified.

Final answer C
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 17 Jul 2026, 23:29
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Given,
Total students = 17
Volunteers in Drama club = 5
Volunteers in Chess Club = 4

To find the number of 3 memeber teams with no two team members represent the same club?

Statement 1:

Robotics = 3
Art = 2

So, adding the volunteers accounted in these clubs = 5+4+3+2 = 14

Remaing 3 volunteers can form teams in ways (1, 1, 1), (1, 2), ( 3)

Not sufficient to have unique answer.

Statement 2:

Exactly one club was represented by fewer than 3 volunteers.

We have teams of 5 & 4, remaining 8 can forms teams in ways
(3, 3, 2), (3, 5), (2, 6), (1, 7)

Not sufficient to have unique answer.

Both statement together:

Teams can be formed in ways (5, 4, 3, 3, 2)

Answer : C (Both statements together are sufficient)

Bunuel
At a school festival, 17 students volunteered to lead activity booths. Each volunteer represented exactly one school club, and the volunteers represented at least 3 clubs. Among the volunteers, 5 represented the Drama Club and 4 represented the Chess Club. If a 3-student team is to be formed so that no two team members represent the same club, in how many ways can the team be formed?

(1) Among the 17 volunteers, 3 represented the Robotics Club and 2 represented the Art Club.
(2) Among the 17 volunteers, exactly one club was represented by fewer than 3 volunteers.


 


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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 17 Jul 2026, 23:34
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t = 17, d = 4, c = 5.
to solve number of ways, we need total number of clubs and student distribution.
S1:
d,c,r,a = 5,4,3,2, remaining = 3 but as we don't know how they are distributed, so not solvable.
S2:
as we no, only one team is fewer than 3.
from remaining 8 it can be, 1,3,4 or 2,3,3. as we don't know the exact distribution. not solvable.
S1&2:
d,c,r,a,x = 5,4,3,2,3
therefore, solvable. C
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 17 Jul 2026, 23:58
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At a school festival, 17 students volunteered to lead activity booths. Each volunteer represented exactly one school club, and the volunteers represented at least 3 clubs. Among the volunteers, 5 represented the Drama Club and 4 represented the Chess Club. If a 3-student team is to be formed so that no two team members represent the same club, in how many ways can the team be formed?

17 students volunteered
one club exactly ; volunteers represented >=3 clubs
Drama = 5
Chess = 4
if 3 student team is to be formed no two team members represent same club , how many ways can team be formed



(1) Among the 17 volunteers, 3 represented the Robotics Club and 2 represented the Art Club.
total students - ( clubs possible for each student )
17- ( 5+4+3+2) ; 3 students to be distributed
the given info is not sufficient to determine teams which can be formed
insufficient

(2) Among the 17 volunteers, exactly one club was represented by fewer than 3 volunteers.

each club had fewer than 3 students , but drama & chess have 5 , 3 students
again this statement is not sufficient ;
insufficient

from statements 1 & 2 together
total students we know 17
3 students can be choosen in 17c3 ways = 680
no 2 team members in same club
we can solve to get teams formed using both statements together

sufficient ; option C is correct
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 18 Jul 2026, 00:43
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Let the club sizes be:
Drama = 5
Chess = 4
Others = ?

We need the number of 3-student teams with all members from different clubs. To answer this, we must know how many volunteers belong to each club.

Statement (1)

Robotics = 3
Art = 2

Now the club sizes are:

Drama = 5
Chess = 4
Robotics = 3
Art = 2

Total so far = 5 + 4 + 3 + 2 = 14

There are 17 volunteers, so 3 volunteers remain.

These 3 volunteers could:

Case 1:
All belong to one new club.

Club sizes:
5, 4, 3, 2, 3

Case 2:
Belong to three different clubs.

Club sizes:
5, 4, 3, 2, 1, 1, 1

The number of valid teams is different in these two cases.

So Statement (1) is not sufficient


Statement (2)

Exactly one club has fewer than 3 volunteers.

We know:
Drama = 5
Chess = 4

The remaining 8 volunteers satisfy the condition, but there are many possibilities.

5, 4, 3, 3, 2

or

5, 4, 5, 1, 2

Different club distributions give different numbers.

So Statement (2) is not sufficient


Statements (1) and (2) together

From (1):

Drama = 5
Chess = 4
Robotics = 3
Art = 2

Three volunteers remain.

From (2):

Exactly one club has fewer than 3 volunteers.

Art already has 2 volunteers so it must be the only club with fewer than 3. Therefore the remaining 3 volunteers must all belong to one new club.

Final club sizes:
Drama = 5
Chess = 4
Robotics = 3
Art = 2
New Club = 3

Now the distribution is fixed so the number of valid teams can be determined uniquely.

Both statements together are sufficient.

Answer: C
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 18 Jul 2026, 01:48
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This is a data sufficiency questions. What we have to remember before starting solving this question is that we do not need the exact answer. Rather we just need to understand if the given info. and statements are sufficient for us to get to a DEFINITE answer.

Now we are given in the stem that 17 students volunteering. These students belong to different school clubs. Each student belongs to exactly one club and there were representatives from atleast 3 clubs. Out of the 17 students, 5 belong to Drama club and 4 belong to the Chess club.
Thus, we have 5+4+other students from other clubs = 17
We have 8 students from other clubs. These students can all be from one club or can be from any number of upto 8 different clubs. What we need to find is the number of ways of choosing a 3 student team, such that none of them belong to the same club. For us to find this, we will need 2 pieces of information:
(a) The exact number of different clubs
(b) The exact number of students in each club

If our statements give us the above 2, we should be able to answer the question definitively. Let's check our statements:

Statement 1: Among the 17 students 3 represented the Robotics Club and 2 represented the Art Club
Thus, we are given 5 from Drama, 4 from Chess, 3 from Robotics and 2 from Art i.e. 14 out of 17 students we know about. For the remaining 3 students, they could be part of one single club or 3 different clubs. Thus there could be students from 5 to 7 different clubs with each club having some unknown number of students. This statement does not give us any information of either of the above two pointers. Hence, we can say it is not sufficient. (A) and (D) eliminated

Statement 2: Among the 17 students exactly one club was represented by fewer than 3 students.
From the stem, we need information about the remaining 8 students. This statement tells us that there is exactly one club with 1 or 2 students. We cannot know if this club has 1 or 2 students. Also, we do not know exactly how many other clubs we have represented and exactly how many students representing each such club. This statement also gives us no information about either of the above pointers. Hence, we can say this statement is not sufficient. We can eliminate (B).

Both Statements Together: From the stem and Statement 1 we know that we have the combination of students as below:
5 Drama, 4 Chess, 3 Robotics and 2 Art = 14 students
We need info. about the remaining 3 students. If you see our analysis of statement 1, we said that we could have 3 students from one club or 1 student each from 3 different clubs or 2 from one club and one from the other. There was no definite way of saying just using statement 1.
But now, when we take both statements together, Statement 2 tells us that exactly one club has fewer than 3 students who represent it. From the above info. we can see that Art is that club. Thus, the only way we can have the remaining 3 students is them being from the same club.
Thus the combination becomes,
5 drama, 4 chess, 3 robotics, 2 art and 3 from one other club.

This answers both the above pointers we needed information on. THus, we can say both statements together are sufficient. (C) is the answer
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 18 Jul 2026, 02:19
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1)
Club sizes are 5, 4, 3, 2 and the remaining 3 volunteers
The remaining 3 could all be from one club (5,4,3,2,3) or from 3 different clubs (5,4,3,2,1,1,1). Different cases give different no. of teams.
Insufficient

2)
We only know there is exactly 1 club with fewer than 3 volunteers.
Club sizes could be (5,4,3,3,2) and (5,4,5,2,1). Different no. of teams
Insufficient

1) + 2)
From (1), the club with 2 volunteers is Art. So the remaining 3 volunteers can't be split up, or we'd have more than 1 club with fewer than 3 volunteers.
Club sizes are fixed as (5,4,3,3,2)
Sufficient

Ans : C
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 18 Jul 2026, 02:41
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lets look at option 1

club sizes known, 5,4,3,2, so this totals 14 students, leaving 3
but those thre could be 5,4,3,2,3 OR 5,4,3,2,1,1,1 for example

so statement 1 is insufficient

statement 2

one club with fewer than three vounteers, remaining eight students beyond drama and chess could be, 7+1, 6+2, 4+3+1....etc
each new distribution will ssatisfy the condition but produce a different number of teams

Both statements:

known club sizes are 5,4,3,2

art club with two volunteeers must be the one that represents fewer than three stuents

there are 3 students remainig so they must all belong to one additional club, as splitting them would violate the rule about only one club having fewwer tham 3 people

5,4,3,2,3

so final c

dont make the mistake of trying to actually workout the total as in DS you dont need that
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remdelectus
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 18 Jul 2026, 03:13
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can i find a unique number of 3 student team, having no two members from the same group.
Clubs:
Drame=5
chess=4
other volunteers=8
Each team must have students from 3 different clubs
Statement 1
Robotics=3
Art=2
club count: 5,4,3,2
3 rem volunteers dist among unknown clubs
they may be distributed in the following ways:
one club of 3(5,4,3,2,3)
three clubs of 1(5,4,3,2,1,1,1)
the number of teams are different so insufficient.

statement 2
Exactly one club is represented by fewer than 3 volunteers.
possibility includes:
(5,4,3,3,2)
(5,4,4,3,1)
Both have a different number of valid teams. Insufficient

Statement 1 and 2 together
Drama=5
chess=4
robotics=3
art=2
rem volunteers=3
from statement 2, exactly one club has fewer than 3 volunteers.
art already has 2v=only club with fewer than 3 volunteers.
so the remaining 3 volunteers must all belong to a single club.
clubs are : 5,4,3,2,3
The number of teams are found so sufficient.
ANS:C.
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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 18 Jul 2026, 03:14
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A alone gives details about 14 students in total, but the 3 students can join a club in any fashion
So Insufficient
B alone - possible ways of distribution can be - 5,4,3,3,2 or 5,4,6,2
Togther upon combining also we see differnet possibilites, So E
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At a school festival, 17 students volunteered to lead activity booths. Each volunteer represented exactly one school club, and the volunteers represented at least 3 clubs. Among the volunteers, 5 represented the Drama Club and 4 represented the Chess Club. If a 3-student team is to be formed so that no two team members represent the same club, in how many ways can the team be formed?

(1) Among the 17 volunteers, 3 represented the Robotics Club and 2 represented the Art Club.
(2) Among the 17 volunteers, exactly one club was represented by fewer than 3 volunteers.


 


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GMAT Club World Cup 2026 (Day 10): At a school festival, 17 students 18 Jul 2026, 03:33
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Total volunteers= 17
Volunteers represented atleast 3 clubs.
D=5 & C=4

1) R=3 & A=2
Remaining volunteers= 17-5-4-3-2=3
The remaining 3 volunteers can be a part of same club or different clubs. It is not sufficient

2) Exactly 1 club has fewer than 3 volunteers
Remaining volunteers = 17-5-4=8
These 8 volunteers can be a part of same club or different clubs. Not enough information. It is not sufficient

Both 1&2
Total Volunteers= 17
D=5 ; C=4 ; R=3 ; A=2
Remaining volunteers can be=3
According to statement two only one club has fewer than 3 members and from statement one we know that is Art Club. So the remaining 3 volunteers must be a part of same club.
We have unique values of numbers of volunteers in each club. With these values we can easily find the teams that can be formed with each representing a different club.
Sufficient

C. Both together are sufficient
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