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In a survey of political preferences, 78% of those asked were in favour of at least one of the proposals: I, II and III. 50% of those asked favoured proposal I, 30% favoured proposal II and 20% favoured proposal III. If 5% of those asked favoured all three of the proposals, what percentage of those asked favoured more than one of the three proposals?

1. 10
2. 12
3. 17
4. 22

Let A=favoured proposal III

B=favoured proposal III

C=favoured proposal III

Here, n(A U B U C)=78

We Knows,

n(AUBUC)= n(A)+n(B)+n(C)-n(A ∩ B)-n(A ∩ C)-n(A ∩ B)+n(A ∩ B ∩ C)

78 = 50 + 30 + 20 - n(A ∩ B) - n(A ∩ C) - n(A ∩ B) + 5.

n(A ∩ B) + n(A ∩ C) + n(A ∩ B) = 27

For those who favored exactly 2 proposals we have to subtract n(A ∩ B ∩ C) from n(A ∩ B) , n(A ∩ C) and n(A ∩ B)

n(A ∩ B) + n(A ∩ C) + n(A ∩ B)  - 3*n(A ∩ B ∩ C)

=27-15 =12

For those who favored exactly 3 proposals=5

Now, those who favored more than one of the 3 proposals = 12 + 5 =17

Hence,Option (C)17 is the correct choice.

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