A={a, b, c, d, e} is given. How many subsets of A contain at least one

Question

A={a, b, c, d, e}  is given. How many subsets of  A  contain at least one vowel?

A. 20 
 
 B. 22 
 
 C. 24 
 
 D. 26 
 
 E. 28

GMATGuruNY · Accepted Answer

Subsets with at least 1 vowel = (all possible subsets) - (subsets with no vowels)

All possible subsets: 
For each of the 5 values in A, there are two options:
to be CHOSEN or NOT CHOSEN.
Since there are 2 options for each of the 5 values, the total number of subsets = 2*2*2*2*2 = 32.

Subset with no vowels :
There are 3 consonants in A:
b, c and d.
For each of these 3 consonants, there are two options:
to be CHOSEN or NOT CHOSEN.
Since there are 2 options for each of the 3 consonants, the total number of subsets with no vowels = 2*2*2 = 8.

Subsets with at least 1 vowel: 
32-8 = 24

C

Mohammadmo · Answer

we have a five member subset(1)
We have 5 four member subsets all of which are among the answer.(5) 
5C3:10 subsets, we need (9)
5C2:10 subsets, we need (7)
5C1:5 subsets, we need (2)
2+7+9+5+1=24
Option C

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MathRevolution · Answer

=>

When we encounter the word “at least” in counting or probability questions, we should consider using complementary counting. We can find the number of outcomes by subtracting the number of complementary outcomes from the total number of outcomes: #total - #complementary.

The complementary outcomes to subsets of A containing at least one vowel are the subsets of A containing only consonants.
The total number of subsets of A is  2^5 = 32 , and the number of subsets of A containing only consonants is  2^3 = 8 .
Thus, the number of subsets of A containing at least one vowel is  32 – 8 = 24 .

Therefore, C is the answer.
Answer: C

satya2029 · Answer

SET- {a,b,c,d,e}
Vowel- {a,e}
Rest- {b,c,d}
For at least one vowel
(2c1*3c1)+(2c1*3c2)+(2c1*3c3)=14
For both the Vowels
(2c2*3c1)+(2c2*3c2)+(2c2*3c3)=7
then count {a,e}, {a} , {e}= 3
Total=14+7+3=24
(C)

Hea234ven · Answer

With the 5 letters in the A set, we can make up the following subsets:
only 1 components- 5C1=5 SUBSETS
With 2 components-5C2= 10 SUBSETS
With 3 components-5C3= 10 SUBSETS
With 4 components-5C4= 5 SUBSETS
With 5 components-5C2= 1 SUBSET
TOTAL= 31 subsets.
However, we are told that at least one components in the subset should be vowel. So, we have to deduct the subsets that include no vowels. By counting the number of possible combinations of b,c,d this is possible. With b,c and d, we can make up a total of 7 subsets (3C1,3C2 AND 3C3).
so the answer would be 31-7=24

JonShukhrat · Answer

Hi Mitch,

Thank you very much for your detailed explanation. According to your explanation, NNNNN (no letter is chosen) or NNN is also a subset of set A. How can they be a subset if they contain no letters? Is it possible for a set to exist without any elements?

Kind regards,

Jon

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A={a, b, c, d, e} is given. How many subsets of A contain at least one

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A={a, b, c, d, e} is given. How many subsets of A contain at least one

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