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# M28-30

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Math Expert
Joined: 02 Sep 2009
Posts: 50627

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16 Sep 2014, 00:29
00:00

Difficulty:

85% (hard)

Question Stats:

42% (00:48) correct 58% (01:27) wrong based on 79 sessions

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How many different subsets of the set {0, 1, 2, 3, 4, 5} do not contain 0?

A. $$16$$
B. $$27$$
C. $$31$$
D. $$32$$
E. $$64$$

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 50627

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16 Sep 2014, 00:29
Official Solution:

How many different subsets of the set {0, 1, 2, 3, 4, 5} do not contain 0?

A. $$16$$
B. $$27$$
C. $$31$$
D. $$32$$
E. $$64$$

Consider the set without 0: {1, 2, 3, 4, 5}. Each out of 5 elements of the set {1, 2, 3, 4, 5} has TWO options: either to be included in the subset or not, so total number of subsets of this set is $$2^5=32$$. Now, each such set will be a subset of {0, 1, 2, 3, 4, 5} and won't include 0.

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24 Sep 2014, 22:30
Hi Bunuel,

In this question should we not subtract 1 from 2^5 to get the answer as 31.
Because that one scenario would mean that none of 1,2,3,4,5 are included in the particular subset.
Math Expert
Joined: 02 Sep 2009
Posts: 50627

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25 Sep 2014, 01:04
2
Hi Bunuel,

In this question should we not subtract 1 from 2^5 to get the answer as 31.
Because that one scenario would mean that none of 1,2,3,4,5 are included in the particular subset.

Set which does not contain any of the terms is an empty set. An empty set is a subset of every non-empty set, so no, we don't have to subtract 1.
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23 Oct 2014, 04:28
4
1
From {1,2,3,4,5}, the sets could be formed as 5C1 + 5C2 + 5C3 + 5C4 + 5C5 + 5C0 ( the empty set). I see what you mean. Thank you!
Intern
Joined: 24 Jun 2015
Posts: 46

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03 Jul 2015, 04:20
Bunuel wrote:
Official Solution:

How many different subsets of the set {0, 1, 2, 3, 4, 5} do not contain 0?

A. $$16$$
B. $$27$$
C. $$31$$
D. $$32$$
E. $$64$$

Consider the set without 0: {1, 2, 3, 4, 5}. Each out of 5 elements of the set {1, 2, 3, 4, 5} has TWO options: either to be included in the subset or not, so total number of subsets of this set is $$2^5=32$$. Now, each such set will be a subset of {0, 1, 2, 3, 4, 5} and won't include 0.

Hi,

I find it hard to understand why it should not be 5¡, because it is suppossed that there are 5 choices that you are combining; I think that answer 2^5=32 seems logic but It is hard to me to visualize why 5¡ does not work here.

Thanks a lot.

Regards

Luis Navarro
Looking for 700
Math Expert
Joined: 02 Sep 2009
Posts: 50627

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03 Jul 2015, 05:12
1
luisnavarro wrote:
Bunuel wrote:
Official Solution:

How many different subsets of the set {0, 1, 2, 3, 4, 5} do not contain 0?

A. $$16$$
B. $$27$$
C. $$31$$
D. $$32$$
E. $$64$$

Consider the set without 0: {1, 2, 3, 4, 5}. Each out of 5 elements of the set {1, 2, 3, 4, 5} has TWO options: either to be included in the subset or not, so total number of subsets of this set is $$2^5=32$$. Now, each such set will be a subset of {0, 1, 2, 3, 4, 5} and won't include 0.

Hi,

I find it hard to understand why it should not be 5¡, because it is suppossed that there are 5 choices that you are combining; I think that answer 2^5=32 seems logic but It is hard to me to visualize why 5¡ does not work here.

Thanks a lot.

Regards

Luis Navarro
Looking for 700

5! gives different arrangements of {0, 1, 2, 3, 4, 5}: {1, 0, 2, 3, 4, 5}, {1, 2, 0, 3, 4, 5}, {1, 2, 3, 0, 4, 5}, ... Which is not what the question is asking.

We need subsets without 0, which are:
{empty};
{1};
...
{5}
{1, 2}
{1, 3}
...
...
{1, 2, 3, 4, 5}
_________________
Intern
Joined: 24 Jun 2015
Posts: 46

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03 Jul 2015, 11:38
Bunuel wrote:
luisnavarro wrote:
Bunuel wrote:
Official Solution:

How many different subsets of the set {0, 1, 2, 3, 4, 5} do not contain 0?

A. $$16$$
B. $$27$$
C. $$31$$
D. $$32$$
E. $$64$$

Consider the set without 0: {1, 2, 3, 4, 5}. Each out of 5 elements of the set {1, 2, 3, 4, 5} has TWO options: either to be included in the subset or not, so total number of subsets of this set is $$2^5=32$$. Now, each such set will be a subset of {0, 1, 2, 3, 4, 5} and won't include 0.

Hi,

I find it hard to understand why it should not be 5¡, because it is suppossed that there are 5 choices that you are combining; I think that answer 2^5=32 seems logic but It is hard to me to visualize why 5¡ does not work here.

Thanks a lot.

Regards

Luis Navarro
Looking for 700

5! gives different arrangements of {0, 1, 2, 3, 4, 5}: {1, 0, 2, 3, 4, 5}, {1, 2, 0, 3, 4, 5}, {1, 2, 3, 0, 4, 5}, ... Which is not what the question is asking.

We need subsets without 0, which are:
{empty};
{1};
...
{5}
{1, 2}
{1, 3}
...
...
{1, 2, 3, 4, 5}

Hi Bunuel,

Sorry, I am a little confused yet, 5! gives different arrangements without 0 {1, 2, 3, 4, 5}, and in your explanation you said:

5! gives different arrangements of {0, 1, 2, 3, 4, 5}: {1, 0, 2, 3, 4, 5}, {1, 2, 0, 3, 4, 5}, {1, 2, 3, 0, 4, 5}, ... Which is not what the question is asking.

But I was reffering to 5! without cero... I am confused, could you help me again?

Thanks a lot

Regards

Luis Navarro
Looking for 700
Math Expert
Joined: 02 Sep 2009
Posts: 50627

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05 Jul 2015, 07:31
1
luisnavarro wrote:
Hi Bunuel,

Sorry, I am a little confused yet, 5! gives different arrangements without 0 {1, 2, 3, 4, 5}, and in your explanation you said:

5! gives different arrangements of {0, 1, 2, 3, 4, 5}: {1, 0, 2, 3, 4, 5}, {1, 2, 0, 3, 4, 5}, {1, 2, 3, 0, 4, 5}, ... Which is not what the question is asking.

But I was reffering to 5! without cero... I am confused, could you help me again?

Thanks a lot

Regards

Luis Navarro
Looking for 700

Yes, 5! gives different arrangements of {1, 2, 3, 4, 5}. 0 in my previous post was a typo.

We need SUBSETS of {0, 1, 2, 3, 4, 5} without 0, not arrangements of {1, 2, 3, 4, 5}. Subsets of {0, 1, 2, 3, 4, 5} without 0 are:
{empty};
{1};
...
{5}
{1, 2}
{1, 3}
...
...
{1, 2, 3, 4, 5}

Notice that arrangements of {1, 2, 3, 4, 5} give the same 5-element sets but arranged in different ways, while subsets give an empty set, 1-element sets, 2-elements set, ..., 5-element sets.

Hope it's clear.
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Joined: 24 Jun 2015
Posts: 46

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05 Jul 2015, 08:03
Bunuel wrote:
luisnavarro wrote:
Hi Bunuel,

Sorry, I am a little confused yet, 5! gives different arrangements without 0 {1, 2, 3, 4, 5}, and in your explanation you said:

5! gives different arrangements of {0, 1, 2, 3, 4, 5}: {1, 0, 2, 3, 4, 5}, {1, 2, 0, 3, 4, 5}, {1, 2, 3, 0, 4, 5}, ... Which is not what the question is asking.

But I was reffering to 5! without cero... I am confused, could you help me again?

Thanks a lot

Regards

Luis Navarro
Looking for 700

Yes, 5! gives different arrangements of {1, 2, 3, 4, 5}. 0 in my previous post was a typo.

We need SUBSETS of {0, 1, 2, 3, 4, 5} without 0, not arrangements of {1, 2, 3, 4, 5}. Subsets of {0, 1, 2, 3, 4, 5} without 0 are:
{empty};
{1};
...
{5}
{1, 2}
{1, 3}
...
...
{1, 2, 3, 4, 5}

Notice that arrangements of {1, 2, 3, 4, 5} give the same 5-element sets but arranged in different ways, while subsets give an empty set, 1-element sets, 2-elements set, ..., 5-element sets.

Hope it's clear.

Thanks, it is more clear know, I only have a little more doubt...

What I inffer for this is that subsets orders by itself "automaticly", so then {1, 2, 3, 4, 5} only consider 1 posible combination, instead of mixing the order in the positions for instance: {3, 1, 5, 4, 2}, {2, 5, 3, 1, 4}... is my conclusion wrong or correct?

Thanks a lot.

Regards.

Luis Navarro
Looking for 700
Math Expert
Joined: 02 Sep 2009
Posts: 50627

### Show Tags

05 Jul 2015, 08:06
2
luisnavarro wrote:
Bunuel wrote:
luisnavarro wrote:
Hi Bunuel,

Sorry, I am a little confused yet, 5! gives different arrangements without 0 {1, 2, 3, 4, 5}, and in your explanation you said:

5! gives different arrangements of {0, 1, 2, 3, 4, 5}: {1, 0, 2, 3, 4, 5}, {1, 2, 0, 3, 4, 5}, {1, 2, 3, 0, 4, 5}, ... Which is not what the question is asking.

But I was reffering to 5! without cero... I am confused, could you help me again?

Thanks a lot

Regards

Luis Navarro
Looking for 700

Yes, 5! gives different arrangements of {1, 2, 3, 4, 5}. 0 in my previous post was a typo.

We need SUBSETS of {0, 1, 2, 3, 4, 5} without 0, not arrangements of {1, 2, 3, 4, 5}. Subsets of {0, 1, 2, 3, 4, 5} without 0 are:
{empty};
{1};
...
{5}
{1, 2}
{1, 3}
...
...
{1, 2, 3, 4, 5}

Notice that arrangements of {1, 2, 3, 4, 5} give the same 5-element sets but arranged in different ways, while subsets give an empty set, 1-element sets, 2-elements set, ..., 5-element sets.

Hope it's clear.

Thanks, it is more clear know, I only have a little more doubt...

What I inffer for this is that subsets orders by itself "automaticly", so then {1, 2, 3, 4, 5} only consider 1 posible combination, instead of mixing the order in the positions for instance: {3, 1, 5, 4, 2}, {2, 5, 3, 1, 4}... is my conclusion wrong or correct?

Thanks a lot.

Regards.

Luis Navarro
Looking for 700

A set, by definition, is a collection of elements without any order. (While, a sequence, by definition, is an ordered list of terms.)

So, we are not interested in ordering the elements in a set (subset): {1, 2} is the same set as {2, 1}.
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Joined: 24 Jun 2015
Posts: 46

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05 Jul 2015, 11:12
Hi Bunuel,

I appreciatte your help, thanks a lot, it is totally clear now.

Best regards.

Luis Navarro
Looking for 700
Intern
Joined: 14 Aug 2015
Posts: 5

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27 Sep 2015, 10:40
Bunuel wrote:
Hi Bunuel,

In this question should we not subtract 1 from 2^5 to get the answer as 31.
Because that one scenario would mean that none of 1,2,3,4,5 are included in the particular subset.

Set which does not contain any of the terms is an empty set. An empty set is a subset of every non-empty set, so no, we don't have to subtract 1.

Hi, Bunuel. Could you please give a link to the rule about empty set being a subset. It sounds strange, taking into account that this term comes from statistics, which never studies 0 resptesentative set. TIA
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Joined: 25 Nov 2015
Posts: 7
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GMAT 1: 660 Q39 V41
GMAT 2: 670 Q39 V42
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WE: Corporate Finance (Real Estate)

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27 Nov 2015, 11:37
kishgau wrote:
From {1,2,3,4,5}, the sets could be formed as 5C1 + 5C2 + 5C3 + 5C4 + 5C5 + 5C0 ( the empty set). I see what you mean. Thank you!

How does this work?

5C1 = 5
5C2 = 10
5C3 = 10
5C4 = 5

There are 30 subsets according to the above. Why would we include 5C5? 5C5 is the entire set, so by definition it is not a subset. Also, why would an empty set be considered a subset?
Manager
Joined: 17 Aug 2015
Posts: 105
GMAT 1: 650 Q49 V29

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09 Sep 2016, 12:38
1
1
seems answer choice D is wrong
subsets of the set {0, 1, 2, 3, 4, 5} do not contain 0?
subset with just one number - 5c1= 5 - (1), (2), (3), (4), (5)
subset with 02 numbers from(1,2,3,4,5) - 5c2 = 10 - example (1,2),(1,3).....
subset with 01 numbers - 5c3= 10 example (1,2,3) (1,2,4)
subset with 04 numbers- 5c4= 5 - example (1,2,3,4), (12,3,5)......
example with 05 numbers 5c5 = 1 the whole one set ( 1,2,3,4,5)

so total = 5+10+10+10+1= 31
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17 Sep 2016, 00:48
Can anyone, please, explain what an empty subset is?
Senior Manager
Joined: 11 Nov 2014
Posts: 334
Location: India
WE: Project Management (Telecommunications)

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17 Sep 2016, 00:54
1
vijaisingh2001 wrote:
seems answer choice D is wrong
subsets of the set {0, 1, 2, 3, 4, 5} do not contain 0?
subset with just one number - 5c1= 5 - (1), (2), (3), (4), (5)
subset with 02 numbers from(1,2,3,4,5) - 5c2 = 10 - example (1,2),(1,3).....
subset with 01 numbers - 5c3= 10 example (1,2,3) (1,2,4)
subset with 04 numbers- 5c4= 5 - example (1,2,3,4), (12,3,5)......
example with 05 numbers 5c5 = 1 the whole one set ( 1,2,3,4,5)

so total = 5+10+10+10+1= 31

Empty subset needs to be considered
Hence 32

Sent from my iPhone
Senior Manager
Joined: 11 Nov 2014
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Location: India
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17 Sep 2016, 00:55
1
Devbek wrote:
Can anyone, please, explain what an empty subset is?

Happy to help

The empty set is the set containing no elements. In mathematics, and more specifically set theory, theempty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

Sent from my iPhone
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17 Sep 2016, 01:39
paidlukkha wrote:
Devbek wrote:
Can anyone, please, explain what an empty subset is?

Happy to help

The empty set is the set containing no elements. In mathematics, and more specifically set theory, theempty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

Sent from my iPhone

Thanks a lot. Math still amazes me with its weird stuff
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05 Jun 2017, 05:25
The other solution to this could be :
Number of ways a subset with only one number can be formed : 5 ways ( taking 1, 2, 3, 4 and 5 each)
Number of ways a subset with two numbers can be formed : 5C2 = 10 ways
Number of ways a subset with three numbers can be formed : 5C3 = 10 ways
Number of ways a subset with four numbers can be formed : 5C4 = 1 way
Null set( empty set) = 1 way

Things to remember while choosing : For example subset (1,2,3) is same as subset (3,2,1) .
Re M28-30 &nbs [#permalink] 05 Jun 2017, 05:25

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# M28-30

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