How many different subsets, including the empty set, of the set {0, 1,

Question

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

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

Bunuel · Accepted Answer

Official Solution:

How many different subsets, including the empty set, 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 of the 5 elements in the set {1, 2, 3, 4, 5} has two possibilities: either to be included in a subset or to be excluded. Therefore, the total number of subsets for this set is \(2^5 = 32\). These 32 subsets include the empty set, single-element subsets like {1}, {2}, {3}, {4}, and {5}, and multi-element subsets like {1, 2}, {1, 2, 3}, {1, 3, 4, 5}, and so on. Each of these subsets will also be a subset of {0, 1, 2, 3, 4, 5} and will not contain the element 0.

Answer: D

Similar question: https://gmatclub.com/forum/fresh-meat-1 ... l#p1215323

Hope it helps.

TGC · Answer

The solution offered counted it differently.

So I just want to clear it up whether I am also thinking in the right direction.

Number of subset

Since we have 5 digits other than 0, we can take any numbers from the set of 5 to make a subset. Also it is a matter of selection and not arrangement.So we will consider combinations.

5c1+5c2+5c3+5c4+5c5=31

And one set is the NULL set having no elements in it so

31+1=32

Please confirm my reasoning of NULL set.

Rgds,
TGC!

KarishmaB · Answer

You want subsets without 0. This means you want all subsets that can be made from {1, 2, 3, 4, 5}. Now for each of the 5 elements, you can choose to include it in your subset or not i.e. there are 2 ways of dealing with each element - include it or don't include it. So there are two choices for each of the 5 elements. So total number of subsets will be = 2*2*2*2*2 = 32 Answer (D) Check: Video on Permutations: https://youtu.be/LFnLKx06EMU Video on Combinations: https://youtu.be/tUPJhcUxllQ Video on Probability: https://youtu.be/0BCqnD2r-kY

samirchaudhary · Answer

Is null set a subset of all the sets ?

Bunuel · Answer

Yes, an empty set is a subset of all sets.

romitsn · Answer

Total number of subsets for a set having n elements=  2^n

So total number of sets with {0, 1, 2, 3, 4, 5} =  2^6 
Without 0, the total number of sets for {1, 2, 3, 4, 5} =  2^5

Hence the answer is  2^6  -  2^5  =  2^5 =32

PerfectScores · Answer

Total number of subsets: 6 + 6C2 + 6C3 + 6C4 + 6C5 + 6C6 = 2^6 - 1 = 64 - 1 = 63

Number of subsets with zero = 1 + 1 x 5C1+ 1 x 5C2 + 1 x 5C3 + 1 x 5C4 = 1 + 5 + 10 + 10 + 5 = 31

Number of subsets without zero = 63 - 31 = 32

godblessme · Answer

The total no of subsets that can be formed is given by  2^n 
Now,
Let us consider a set without 0  (1,2,3,4,5)  in total there are 5 elements so possible no of sets with these 5 elements is  2^5=32 
Hence
There are 32 sets which do not contain the number 0.

griffith7 · Answer

I did not know the subset formula when trying to solve this initially - clearly it is a very simple and easy to remember formula that renders the problem easy at a mechanical level, if you understand the formula - but I am struggling to see why the alternative logic I used initially produced an incorrect answer. Can anyone explain what element is missing?

Sets consisting of 6 numbers: 0 (because one of the numbers must be excluded)
Sets consisting of 5 numbers: 1 (1,2,3,4,5)
Sets consisting of 4 numbers: 5 (combination formula; 5 choose 4)
Sets consisting of 3 numbers: 10 (combination formula; 5 choose 3)
Sets consisting of 2 numbers: 10 (combination formula; 5 chose 2)
Sets consisting of 1 number: 5 (1 or 2 or 3 or 4 or 5)

All of which sums to 31. What subset is not being picked up?

Bunuel · Answer

You are missing an empty set, which is a subset of all sets.

onlyPlanA · Answer

So, if we want the number of different subsets, including the empty set, that do not contain 0 and 1 both are:

2^4  or  2^6 - 2^4  ???

Bunuel · Answer

If the subsets cannot contain either 0 or 1, the answer is 2^4.

If they may contain 0 or 1, but not both together, the answer is 2^6 - 2^4.

dhart · Answer

Set= {0,1,2,3,4,5}
we don't the element 0 to be included in any of the subsets 
os, new set = {1,2,3,4,5} this subset excludes element 0
total subsets for new set = 2^5 =32 
32 subsets include {},{1}.....{1,2},.....{1,2,3}......upto {1,2,3,4,5}

and these 32 subsets are also subsets of original set {0,1,2,3,4,5}

option D

icandoit4u · Answer

so the set is {0,1,2,3,4,5}
we need total subsets without 0, so i just took out 0 and solved it as a set of 5 numbers, 
first number of 1 digit set = eg (1), (2), (3), (4), (5) this is 5 in total = 5 
2 digit set for eg (1,2),(2,3)= so total no of ways is using nCr = 5C2=  5 
3 digit set for eg (1,2,3) = total no of way= 5C3=  10 
4 digit set for eg (1,2,3,4) = total no of ways = 5C4=  10 
5 digit set = 5C5= 1 
1 empty set {}= 1 
total =  5+5+10+10+1+1=32  
 ans 32

How many different subsets, including the empty set, of the set {0, 1,

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How many different subsets, including the empty set, of the set {0, 1,

Prep Toolkit

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