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Sets - counting subsets

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Sets - counting subsets

by snigdha1605 » Mon Jul 29, 2013 4:10 am
How many different subsets of the set {0, 1, 2, 3, 4, 5} do not contain 0?

16
27
31
32
64

OA- 32

[spoiler]Can someone help me with the below logic -
Total sets without 0 =
5 (Individual sets of 1 number each - {1},{2}, etc)
5C2 (Sets of 2 numbers each)
5C3 (sets of 3 numbers each)
5C4 (sets of 4 numbers each)
5C5 (Set of 5 numbers)

= 5+10+10+5+1 = 31

(which set am i missing?)

[/spoiler][/spoiler]
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by GMATGuruNY » Mon Jul 29, 2013 4:58 am
snigdha1605 wrote:How many different subsets of the set {0, 1, 2, 3, 4, 5} do not contain 0?

16
27
31
32
64

OA- 32
Any of the following values may be included in a subset:
1, 2, 3, 4, 5
For each of these values, there are 2 options: TO INCLUDE the value or NOT TO INCLUDE the value.
Since there are 5 values, and 2 options for each value, the total number of possible subsets = 2*2*2*2*2 = 32.

The correct answer is D.

The 32 options above include the EMPTY SET: the case in which NONE of the 5 values is selected.
The EMPTY SET is considered a subset of EVERY SET.
This concept is beyond the scope of the GMAT, so please feel free to ignore this problem.
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by snigdha1605 » Mon Jul 29, 2013 5:20 am
Thanks Mitch!
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by Brent@GMATPrepNow » Mon Jul 29, 2013 5:54 am
Hi snigdha1605,

I thought I'd point out that your approach of adding 5C1 + 5C2 + 5C3 etc would have worked (except you missed 5C0). Once you get to the point where you are adding all possible combinations, you can use a nice rule that says:
nC0 + nC1 + nC2 + . . . . nCn = 2^n
So, for example, 5C0 + 5C1 + 5C2 + 5C3 + 5C4 + 5C5 = 2^5
Similarly, 7C0 + 7C1 + 7C2 + . . . 7C6 + 7C7 = 2^7

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by amey90 » Fri May 30, 2014 3:03 am
@Brent, what doe 5C0 signify here?

5C5 means you select 5 nos from 5 or 5C1 means you select 1 no. from 5 nos.
wouldn't 5C0 would just mean a blank set?

Thanks.
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by Brent@GMATPrepNow » Fri May 30, 2014 7:27 am
amey90 wrote:@Brent, what doe 5C0 signify here?

5C5 means you select 5 nos from 5 or 5C1 means you select 1 no. from 5 nos.
wouldn't 5C0 would just mean a blank set?

Thanks.
5C0 represents the number of ways we can select 0 objects from a set of 5 objects. There is 1 way to accomplish this. In other words, 5C0 = 1

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