Last visit was: 19 Apr 2025, 07:32 It is currently 19 Apr 2025, 07:32
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Close
Request Expert Reply
Confirm Cancel
User avatar
gmatt1476
Joined: 04 Sep 2017
Last visit: 27 Mar 2025
Posts: 334
Own Kudos:
Given Kudos: 62
Posts: 334
Kudos: 23,539
 [220]
8
Kudos
Add Kudos
212
Bookmarks
Bookmark this Post
Most Helpful Reply
User avatar
CareerGeek
Joined: 20 Jul 2017
Last visit: 18 Apr 2025
Posts: 1,296
Own Kudos:
3,965
 [63]
Given Kudos: 162
Location: India
Concentration: Entrepreneurship, Marketing
GMAT 1: 690 Q51 V30
WE:Education (Education)
GMAT 1: 690 Q51 V30
Posts: 1,296
Kudos: 3,965
 [63]
33
Kudos
Add Kudos
29
Bookmarks
Bookmark this Post
User avatar
CareerGeek
Joined: 20 Jul 2017
Last visit: 18 Apr 2025
Posts: 1,296
Own Kudos:
3,965
 [27]
Given Kudos: 162
Location: India
Concentration: Entrepreneurship, Marketing
GMAT 1: 690 Q51 V30
WE:Education (Education)
GMAT 1: 690 Q51 V30
Posts: 1,296
Kudos: 3,965
 [27]
19
Kudos
Add Kudos
8
Bookmarks
Bookmark this Post
General Discussion
User avatar
nick1816
User avatar
Retired Moderator
Joined: 19 Oct 2018
Last visit: 18 Apr 2025
Posts: 1,854
Own Kudos:
7,537
 [7]
Given Kudos: 707
Location: India
Posts: 1,854
Kudos: 7,537
 [7]
6
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Assume the elements in set A= n

Statement 1-
Number of ways to select 2 elements out of n is 6
nC2=6
We can figure out n

Sufficient

Statement 2
nC0+ nC1+nC2+......+nCn=16
\(2^n\)=16

OR

Any element can either present in the subset or not.
Hence total number of subsets- 2*2*2...n times= \(2^n\)

Anyways, we can figure out 'n'

Sufficient



gmatt1476
The cardinality of a finite set is the number of elements in the set. What is the cardinality of set A ?

(1) 2 is the cardinality of exactly 6 subsets of set A.
(2) Set A has a total of 16 subsets, including the empty set and set A itself.


DS06351.01
avatar
germanandres82
Joined: 20 May 2019
Last visit: 26 Nov 2019
Posts: 2
Own Kudos:
18
 [16]
Given Kudos: 3
Posts: 2
Kudos: 18
 [16]
16
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Dillesh4096
gmatt1476
The cardinality of a finite set is the number of elements in the set. What is the cardinality of set A ?

(1) 2 is the cardinality of exactly 6 subsets of set A.
(2) Set A has a total of 16 subsets, including the empty set and set A itself.


DS06351.01

Let 'n' be the cardinality of set A.

(1) 2 is the cardinality of exactly 6 subsets of set A.
Number of 2 sets that can be formed from a set of n elements = nc2 = 6
--> n(n-1)/2 = 6
--> n(n-1) = 12 = 4*3
By comparison, n = 4
So, Cardinality of set A = 4 -- Sufficient


Formula: Number of subsets possible for a set of n elements = \(2^n\) (including the empty set & the set itself)
E.g: If Set A = {1, 2, 3}, All subsets possible = {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}, {} = 8 subsets i.e, \(2^3\)


(2) Set A has a total of 16 subsets, including the empty set and set A itself.
--> \(2^n\) = 16
--> n = 4
So, Cardinality of set A = 4 -- Sufficient

IMO Option D

I am honestly completely lost on this one, seems that I have a completely different idea on what the problem is asking.

When I read "cardinality", I understand basically the number of elements in a set. Following this, my interpretation of the question is if I can know the number of elements in set A.

1) 2 is the cardinality of exactly 6 subsets of set A.

There are 6 subsets in set A with 2 elements INSUFFICIENT - THERE COULD BE OTHER SUBSETS IN A OR EVEN SOME OF THE ELEMENTS CAN BELONG TO MULTIPLE SUBSETS

II) Set A has a total of 16 subsets, including the empty set and set A itself INSUFFICIENT - NO INFORMATION ON NUMBER OF ELEMENTS IN SET A

I)-II) INSUFFICIENT - WE KNOW THAT 6 OUT OF THE 16 SUBSETS OF SET A HAVE " ELEMENTS; YET WE DO NOT KNOW HOW MANY ELEMENTS THE OTHER 10 SUBSETS HAVE NOR IF SOME ELEMENTS ARE IN MORE THAN 1 SUBSET

Clearly the question is going in a completely different direction than my line of though, can someone PLEASE help me explain where do I get completely lost?

Thank you!!!
User avatar
Camach700
Joined: 07 Jan 2019
Last visit: 02 Feb 2023
Posts: 44
Own Kudos:
71
 [5]
Given Kudos: 62
GMAT 1: 710 Q48 V38
GPA: 3.78
GMAT 1: 710 Q48 V38
Posts: 44
Kudos: 71
 [5]
3
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
germanandres82

First of all, I believe the first statement could have been structured in a clearer way. First statement actually says: "The number of 2 cardinality subsets in set A is exactly 6", which means there are exactly (and only) 6 two-cardinality subsets in A (no more, no less).

That being said, we can analyze the two statements:

(1) Sufficient. If there are exactly 6 subsets which have cardinality = 2 in set A that means we can make exactly 6 groups of 2 elements out of set A.

Let n be the number of elements in set A. C(n¦2) = n(n-1)(n-2)!/(n-2)!(2!) = 6; from here we have n = 4

(2) Sufficient. Set A has exactly 16 subsets (no more, no less). That means we can make exactly 16 different combinations out of n.

#of combinations we can take out of any set A is: C(n¦0) + C(n¦1) +...+ C(n¦n) = 16. Which n satisfies this condition?

Let's try with n = 1 --> C(1¦0) + C(1¦1) = 2. Nope.

Let's try with n = 2 --> C(2¦0) + C(2¦1) + C(2¦2) = 4. Nope.

Also note that ∑C(n¦k), [with k=0,1,2,...,n] equals 2^n.

Therefore we have: 16 = 2^n from which n=4



Hope that helps
avatar
Gmat20201
Joined: 25 Jan 2020
Last visit: 01 May 2021
Posts: 34
Own Kudos:
32
 [4]
Given Kudos: 799
Location: United Arab Emirates
Concentration: General Management, Operations
GPA: 3.1
Products:
Posts: 34
Kudos: 32
 [4]
4
Kudos
Add Kudos
Bookmarks
Bookmark this Post
The cardinality of a finite set is the number of elements in the set. What is the cardinality of set A ?

(1) 2 is the cardinality of exactly 6 subsets of set A.
(2) Set A has a total of 16 subsets, including the empty set and set A itself.

Soln:

Question simply is asking us - How many elements are there in Set A

Lets check each statement one by one.


St(1) 2 is the cardinality of exactly 6 subsets of set A.

Simplifying statement for understanding- st 1 says that - 6 subsets of set A can be formed picking two elements

For example - If there is set A with 3 elements (1,2,3)- i can have subset with no element, second subset with 1 element each , third subset with 2 elements and fourth set of subset with 3 elements in it.

3C0+3C1+3C2+3C3= Total Subset of set A

So for the given set in Question ,lets say it has n elements
St 1 gives us information - that nC2=6

We need to find n here..

To cal n here lets solve
nC2= n!/2! (n-2)!
= n(n-1) (n-2)!/2! (n-2)!

Simplifying can i write n! as n (n-1) (n-2)! [ for understanding- if n=6 ,than 6! can be written as 6*5*4!--same way]

= n(n-1)/2! [(n-2)! cancels out]
=n(n-1)/2 [ we know 2! =2*1=2]

St 1 says nC2= 6
n(n-1)/2= 6

Solving further n(n-1)=12
n^2-n-12=0
n=4 or n=-3

obviously it cannot be -3 , so we know n=4.. Set A has 4 elements. St 1 is sufficient.



(2) Set A has a total of 16 subsets, including the empty set and set A itself.

Total subsets=16 (given)

Remember-Number of subsets for a set of n elements= 2^n

We know 2^n=16
n=4

Sufficient.

Ans is D

Hope this detailed explanation helps!
User avatar
KarishmaB
Joined: 16 Oct 2010
Last visit: 18 Apr 2025
Posts: 15,889
Own Kudos:
72,691
 [4]
Given Kudos: 462
Location: Pune, India
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 15,889
Kudos: 72,691
 [4]
4
Kudos
Add Kudos
Bookmarks
Bookmark this Post
gmatt1476
The cardinality of a finite set is the number of elements in the set. What is the cardinality of set A ?

(1) 2 is the cardinality of exactly 6 subsets of set A.
(2) Set A has a total of 16 subsets, including the empty set and set A itself.


DS06351.01

This is a P&C question masquerading as a Sets question. Perhaps that is why it seems confusing.

Cardinality = Number of elements in the set

So say set A has n elements. Then its cardinality is n.
We need to find the number of elements in set A.

(1) 2 is the cardinality of exactly 6 subsets of set A.

When, from set A, we make all subsets, exactly 6 subsets have 2 elements each. Think about it - if you have a set with n elements. How will you make subsets with exactly 2 elements? How many such distinct subsets can you make? In nC2 ways.
So we are given that nC2 = 6
Then n must be 4 because 4C2 = 6. For all greater values of n, nC2 will be greater.
Sufficient alone.

(2) Set A has a total of 16 subsets, including the empty set and set A itself.

From the n elements of set A, total how many subsets can you make? For each element, you can either select it or ignore it. So each element can he handled in 2 ways. IF w have n elements, we get 2*2*2*.. n times
We are given that 2^n = 16
n = 4
Sufficient alone.

Answer (D)

Take an example to help see it clearly. Say set A = {5, 7, 8, 9}
How many subsets can you make with exactly 2 elements? 4C2 = 6. They will include {5, 7}, {5, 8}, {5, 9}, {7, 8}, {7, 9} and {8, 9}
How many total subsets can you make? You can take the 5 or ignore. You can take the 7 or ignore etc. You have subsets ranging from {}, {5}, {7}, ... till {5, 7, 8, 9}
(This is similar to finding total number of factors for a positive integer)
User avatar
Tanchat
Joined: 31 Jan 2020
Last visit: 20 Jun 2023
Posts: 226
Own Kudos:
Given Kudos: 139
Posts: 226
Kudos: 17
Kudos
Add Kudos
Bookmarks
Bookmark this Post
KarishmaB
gmatt1476
The cardinality of a finite set is the number of elements in the set. What is the cardinality of set A ?

(1) 2 is the cardinality of exactly 6 subsets of set A.
(2) Set A has a total of 16 subsets, including the empty set and set A itself.


DS06351.01

This is a P&C question masquerading as a Sets question. Perhaps that is why it seems confusing.

Cardinality = Number of elements in the set

So say set A has n elements. Then its cardinality is n.
We need to find the number of elements in set A.

(1) 2 is the cardinality of exactly 6 subsets of set A.

When, from set A, we make all subsets, exactly 6 subsets have 2 elements each. Think about it - if you have a set with n elements. How will you make subsets with exactly 2 elements? How many such distinct subsets can you make? In nC2 ways.
So we are given that nC2 = 6
Then n must be 4 because 4C2 = 6. For all greater values of n, nC2 will be greater.
Sufficient alone.

(2) Set A has a total of 16 subsets, including the empty set and set A itself.

From the n elements of set A, total how many subsets can you make? For each element, you can either select it or ignore it. So each element can he handled in 2 ways. IF w have n elements, we get 2*2*2*.. n times
We are given that 2^n = 16
n = 4
Sufficient alone.

Answer (D)

Take an example to help see it clearly. Say set A = {5, 7, 8, 9}
How many subsets can you make with exactly 2 elements? 4C2 = 6. They will include {5, 7}, {5, 8}, {5, 9}, {7, 8}, {7, 9} and {8, 9}
How many total subsets can you make? You can take the 5 or ignore. You can take the 7 or ignore etc. You have subsets ranging from {}, {5}, {7}, ... till {5, 7, 8, 9}
(This is similar to finding total number of factors for a positive integer)

KarishmaB

The word "exactly" tells us that there are "only" 6 subsets (of A) that have 2 elements, right?

If there is no "exactly" in Statement (1). There could be other subsets of A and Statement (1) will be insufficient.
User avatar
KarishmaB
Joined: 16 Oct 2010
Last visit: 18 Apr 2025
Posts: 15,889
Own Kudos:
72,691
 [2]
Given Kudos: 462
Location: Pune, India
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 15,889
Kudos: 72,691
 [2]
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Tanchat
KarishmaB
gmatt1476
The cardinality of a finite set is the number of elements in the set. What is the cardinality of set A ?

(1) 2 is the cardinality of exactly 6 subsets of set A.
(2) Set A has a total of 16 subsets, including the empty set and set A itself.


DS06351.01

This is a P&C question masquerading as a Sets question. Perhaps that is why it seems confusing.

Cardinality = Number of elements in the set

So say set A has n elements. Then its cardinality is n.
We need to find the number of elements in set A.

(1) 2 is the cardinality of exactly 6 subsets of set A.

When, from set A, we make all subsets, exactly 6 subsets have 2 elements each. Think about it - if you have a set with n elements. How will you make subsets with exactly 2 elements? How many such distinct subsets can you make? In nC2 ways.
So we are given that nC2 = 6
Then n must be 4 because 4C2 = 6. For all greater values of n, nC2 will be greater.
Sufficient alone.

(2) Set A has a total of 16 subsets, including the empty set and set A itself.

From the n elements of set A, total how many subsets can you make? For each element, you can either select it or ignore it. So each element can he handled in 2 ways. IF w have n elements, we get 2*2*2*.. n times
We are given that 2^n = 16
n = 4
Sufficient alone.

Answer (D)

Take an example to help see it clearly. Say set A = {5, 7, 8, 9}
How many subsets can you make with exactly 2 elements? 4C2 = 6. They will include {5, 7}, {5, 8}, {5, 9}, {7, 8}, {7, 9} and {8, 9}
How many total subsets can you make? You can take the 5 or ignore. You can take the 7 or ignore etc. You have subsets ranging from {}, {5}, {7}, ... till {5, 7, 8, 9}
(This is similar to finding total number of factors for a positive integer)

KarishmaB

The word "exactly" tells us that there are "only" 6 subsets (of A) that have 2 elements, right?

If there is no "exactly" in Statement (1). There could be other subsets of A and Statement (1) will be insufficient.

Not really. "Exactly" only makes the situation crystal clear. It clarifies that when you take all subsets into account, exactly 6 have cardinality 2.
Even if we were given "2 is the cardinality of 6 subsets of set A," we would assume that of the total number of subsets of A, 6 have cardinality 2. This implies that others do not have cardinality 2.

Take another example: 6 of the team members have tested positive.
What does this mean? That others have not. Otherwise, we would have included them too.
If we were awaiting results for others, we would have said "6 of the team members have already tested positive."
When we give an exact number, it pretty much implies it is the EXACT number.
User avatar
Rickooreo
Joined: 24 Dec 2021
Last visit: 15 Feb 2023
Posts: 305
Own Kudos:
Given Kudos: 240
Location: India
Concentration: Finance, General Management
GMAT 1: 690 Q48 V35
GPA: 3.95
WE:Real Estate (Consulting)
GMAT 1: 690 Q48 V35
Posts: 305
Kudos: 29
Kudos
Add Kudos
Bookmarks
Bookmark this Post
gmatophobia

Hi,
Can you please help me understand what cardinality is?
Basis what is mentioned in the question, I interpreted it as follows
A = {1,2,3} thus cardinality is 3.
Assuming this as correct interpretation

Statement : 1) 2 is the cardinality of exactly 6 subsets of set A.

There are 6 subsets in set A with 2 elements
Insufficient - There could be other subsets in A or even some of the elements can belong to multiple subsets i.e subsets having cardinality more than or less than 2.

Also, 2 is the cardinality of exactly 6 subsets of set A, I understood it as there will be total of 12 elements with 6 pairs each having 2 elements each
User avatar
gmatophobia
User avatar
Quant Chat Moderator
Joined: 22 Dec 2016
Last visit: 18 Apr 2025
Posts: 3,114
Own Kudos:
8,189
 [1]
Given Kudos: 1,860
Location: India
Concentration: Strategy, Leadership
Products:
Posts: 3,114
Kudos: 8,189
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Rickooreo
gmatophobia

Hi,
Can you please help me understand what cardinality is?
Basis what is mentioned in the question, I interpreted it as follows
A = {1,2,3} thus cardinality is 3.
Assuming this as correct interpretation

Statement : 1) 2 is the cardinality of exactly 6 subsets of set A.

There are 6 subsets in set A with 2 elements
Insufficient - There could be other subsets in A or even some of the elements can belong to multiple subsets i.e subsets having cardinality more than or less than 2.

Also, 2 is the cardinality of exactly 6 subsets of set A, I understood it as there will be total of 12 elements with 6 pairs each having 2 elements each

Rickooreo - You have correctly identified the definition of cardinality.

There are 6 subsets in set A with 2 elements

This means if we select any two elements in the set, there are 6 unique subsets.

There may be other subsets with cardinality 3, with cardinality 4 so on. But with cardinality 2, only 6 sub set exists.

In terms of P&C - The statement is similar to saying "the number of ways of selecting two numbers from a set which has n numbers" is 6.

nC2 = 6

n = 4
User avatar
Kavicogsci
Joined: 13 Jul 2024
Last visit: 09 Feb 2025
Posts: 174
Own Kudos:
Given Kudos: 154
GMAT 1: 710 Q48 V40
GMAT 1: 710 Q48 V40
Posts: 174
Kudos: 65
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hi Bunuel KarishmaB not clear how we get to total subsets by saying every element can be chosen/not chosen hence total possiblites of n elements is 2*2*2...n = 2^n
Can you give any other combinations problem where we may be doing this. I am not getting intuitively why we are multiplying even though it sounds very practical. I want to make sure I truly get it and not just memorize the formula.

I was so fixated with
Nc0+Nc1 + Nc2 +....+ NcN = Sum of subsets so i thought we couldn't find it
Moderator:
Math Expert
100761 posts