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

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

by Max@Math Revolution » Tue Aug 27, 2019 1:08 am

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[GMAT math practice 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
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by GMATGuruNY » Tue Aug 27, 2019 3:01 am
Max@Math Revolution wrote:[GMAT math practice 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
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

The correct answer is C.
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by Max@Math Revolution » Thu Aug 29, 2019 8:48 am
=>

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
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