In a political survey, 78% of the politicians favour at least one proposal, 50% of them are in favour of proposal A, 30% are in favour of proposal B, Band 20% are in favour of proposal C. 5% are in favour of all three proposals. what is the percentage of people favouring more than one proposal?

Options

A. 16
B. 17
C. 18
D. 19

Answer

To solve the problem, we will use the principle of inclusion-exclusion to find the percentage of politicians who favor more than one proposal.

### Step 1: Define the variables
Let:
- \( n(A) \) = Percentage of politicians in favor of proposal A = 50%
- \( n(B) \) = Percentage of politicians in favor of proposal B = 30%
- \( n(C) \) = Percentage of politicians in favor of proposal C = 20%
- \( n(A \cap B \cap C) \) = Percentage of politicians in favor of all three proposals = 5%
- \( n(A \cup B \cup C) \) = Percentage of politicians in favor of at least one proposal = 78%

### Step 2: Use the inclusion-exclusion principle
According to the inclusion-exclusion principle:
\[ n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C) \]

### Step 3: Rearrange to find the intersections
We need to find the total percentage of politicians who favor more than one proposal. We can express the intersections as:
\[ n(A \cap B) + n(A \cap C) + n(B \cap C) = n(A) + n(B) + n(C) + n(A \cap B \cap C) - n(A \cup B \cup C) \]

### Step 4: Substitute known values
Substituting the known values into the equation:
\[ n(A \cap B) + n(A \cap C) + n(B \cap C) = 50 + 30 + 20 + 5 - 78 \]
\[ n(A \cap B) + n(A \cap C) + n(B \cap C) = 105 - 78 = 27 \]

### Step 5: Find the percentage of people favoring more than one proposal
To find the percentage of people favoring more than one proposal, we need to subtract the percentages of those who favor only one proposal from the total who favor at least one proposal.

Let \( x \) be the percentage of people who favor only one proposal:
\[ x = n(A) + n(B) + n(C) - 2(n(A \cap B) + n(A \cap C) + n(B \cap C)) + 3n(A \cap B \cap C) \]
This means:
\[ x = 50 + 30 + 20 - 2 \cdot 27 + 3 \cdot 5 \]
\[ x = 100 - 54 + 15 = 61 \]

### Step 6: Calculate the percentage favoring more than one proposal
Now, we can find the percentage of people favoring more than one proposal:
\[ \text{Percentage favoring more than one proposal} = n(A \cup B \cup C) - x \]
\[ = 78 - 61 = 17 \]

### Final Answer
The percentage of people favoring more than one proposal is **17%**.

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