A sample poll of 200 votes revealed the following information concerning three candidates A, B and C of a certain party who were running for three different offices.
28 in favour of both A and B.
98 in favour of A or B but not C.
42 in favour of B but not A or C.
122 in favour of B or C but not A.
64 in favour of C but not Aor B.
14 in favour of A and C but not B.
How many voters were in favour of all the three candidates?
How many voters were in favour of all the three candidates?
- 14
- 8
- 20
- 16
The Correct Option is B
Solution and Explanation
From the question we know that,
28 in favor of both A and B.
98 in favor of A or B but not C.
42 in favor of B but not A or C.
122 in favor of B or C but not A.
64 in favor of C but not A or B.
14 in favor of A and C but not B.
So we will denote the following as
Number of voters in favor of candidate A. = \(|A|\)
Number of voters in favor of candidate B. = \(|B|\)
Number of voters in favor of candidate C. = \(|C|\)
Number of voters in favor of both A and B. = \(|A ∩ B|\)
Number of voters in favor of both A and C. = \(|A ∩ C|\)
Number of voters in favor of both B and C. = \(|B ∩ C|\)
Number of voters in favor of all three candidates. = \(|A ∩ B ∩ C|\)
Assume that,
\(|A ∩ B ∩ C| = x\)
\(|A ∩ B ∩ C| = 28 - x\)
\(|A ∩ C ∩ B^| = 14\)
We know that,
\(|(A ∪ B) ∩ C| = |A ∩ B ∩ C| + |A ∩ C ∩ B | + |B ∩ A ∩ C|\)
Putting the values in the equation,
\(98 = (28 - x) + |A ∩ C ∩ B | + 42\)
\(98 = 28 - x + |A ∩ C ∩ B| + 42\)
\(|A ∩ C ∩ B| = 28 +x\)
For \(|(B ∪ C) ∩ A|\)
\(|(B ∪ C) ∩ A| = |B ∩ C ∩ A| + |B ∩ A ∩ C| + |C ∩ A ∩ B|\)
Putting the values in the equation,
\(122 = |B ∩ C ∩ A| + 42 + 64\)
\(|B ∩ C ∩ A| = 16\)
Now putting the values in final equation
\(|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|\)
= 8
The correct option is (B): 8
How many voters were in favour of A irrespective of B or C?
How many voters were in favour of A irrespective of B or C?
- 78
- 64
- 42
- 56
The Correct Option is A
Solution and Explanation
For voters in favor of \(A\)
\(|A| = |A ∩ B ∩ C| + |A ∩ B| + |A ∩ C| + |A ∩ B ∩ C|\) – (eq)
\(\)Here,
\(|A ∩ B ∩ C|\) = voters favoring all candidates
\(|A ∩ B|\) = 28 (Voters in favor of both \(A\) and \(B\)
\(|A ∩ C|\) = 14 (Voters in favor of both \(A\) and \(C\)
Putting values in above equation,
= 78
The correct option is (A): 78
How many voters werein favour of B irrespective of A or C?
- 78
- 62
- 48
- 86
The Correct Option is D
Solution and Explanation
For voters in favor of B irrespective of A or C
Voters in favor of B only = \(∣B∩A'∩C'∣\) = 42
Voters in favor of both B and C but not A = \(∣B∩C∩A'∣\) = 16
Voters in favor of both B and A but not C = \(∣A∩B∩C'∣\) = 20
Voters in favor of all three = \(∣A∩B∩C∣\) = 8
\(∣B∣=∣B∩A'∩C'∣+∣B∩C∩A'∣+∣A∩B∩C'∣+∣A∩B∩C∣\)
\(∣B∣ = ∣B∩A'∩C'∣ + ∣B∩C∩A'∣ + ∣A∩B∩C'∣ + ∣A∩B∩C∣\)
\(|B|\) = 42 + 16 + 20 + 8 = 86
The voters in favor of \(B\) irrespective of \(A\) or \(C\) are 86
The correct option is (D): 86
How many voters were in favour of C irrespective of A or B?
- 78
- 102
- 88
- 86
The Correct Option is B
Solution and Explanation
\(∣C∣ = ∣C∩A'∩B'∣+∣A∩C∩B'∣+∣B∩C∩A'∣+∣A∩B∩C∣\)
= \(∣C∣ = 64+14+16+8=102\)
The voters in favor of \(C\) irrespective of \(A\) or \(B\) are 102
The correct option is (B): 102
How many voters werein favour of A and B but not C?
- 8
- 20
- 14
- 16
The Correct Option is B
Solution and Explanation
From the above information, we know that
\(∣A∩B∩C'∣\) = \(28 - x\)
\(x\) = 8 as we have solved it above
28 - 8
Voters were in favor of A and B but not C= 20
The correct option is (B): 20
How many voters werein favour of only one of the candidates ?
- 58
- 78
- 106
- 142
The Correct Option is D
Solution and Explanation
From the above, we know that
\(∣A∩B'∩C'∣=28+x=36\)
\(∣B∩A'∩C'∣=42\)
\(∣C∩A'∩B'∣=64\)
Voters were in favor of only one of the candidates = 36 + 42 + 64
= 142
The correct option is (D): 142
How many voters werein favour of A and C but not B?
- 22
- 14
- 16
- 20
The Correct Option is B
Solution and Explanation
It is given in the question that 14 voters are in favor of A and C but not B
The correct option is (B): 14
How many voters were in favour of C alone?
- 36
- 42
- 64
- 38
The Correct Option is C
Solution and Explanation
It is given in the question that 64 voters are in favor of C alone
The correct option is (C): 64
How many voters were in favour of B and C but not A?
- 16
- 14
- 42
- 64
The Correct Option is A
Solution and Explanation
It is given in the question that 16 voters are in favor of B and C but not A
The correct option is (A): 16
How many voters were in favour of A and C but not B?
- 16
- 14
- 36
- 42
The Correct Option is B
Solution and Explanation
It is given in the question that 14 voters are in favor of A and C but not B
The correct option is (B): 14
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