Correct Option (C)
The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
To determine the total number of persons who attended the party, we first establish the foundational relationships from the question stem:
- Number of persons who took tea, |T| = 75.
- Number of persons who took coffee, |C| = 60.
- Number of persons who took both tea and coffee, |T ∩ C| = 15.
- No one taking milk takes tea, implying |M ∩ T| = 0. Consequently, |T ∩ C ∩ M| = 0.
- Each person takes at least one drink.
The number of persons who took tea or coffee is: |T ∪ C| = |T| + |C| - |T ∩ C| = 75 + 60 - 15 = 120.
The total number of persons who attended the party, |P|, can be expressed using the principle of inclusion-exclusion for three sets (Tea, Coffee, Milk): |P| = |T| + |C| + |M| - (|T ∩ C| + |T ∩ M| + |C ∩ M|) + |T ∩ C ∩ M|
Substituting the given values: |P| = 75 + 60 + |M| - (15 + 0 + |C ∩ M|) + 0 |P| = 135 + |M| - 15 - |C ∩ M| |P| = 120 + |M| - |C ∩ M|
This equation, |P| = 120 + |M| - |C ∩ M|, is the basis for evaluating the sufficiency of the statements.
Evaluation of Statement-1 alone:
Statement-1 provides: Number of persons who took milk, |M| = 50.
Substituting |M| = 50 into the fundamental equation: |P| = 120 + 50 - |C ∩ M| = 170 - |C ∩ M|.
Since the number of persons who took both coffee and milk (|C ∩ M|) is unknown, the total number of persons (|P|) cannot be uniquely determined. Therefore, Statement-1 alone is insufficient.
Evaluation of Statement-2 alone:
Statement-2 provides: The number of persons who attended the party, |P|, is five times the number of persons who took milk only.
Let |M_only| represent persons who took only milk. Given |M ∩ T| = 0, persons who took only milk are those who took milk but neither tea nor coffee. Thus, |M_only| = |M| - |C ∩ M|.
From Statement-2, |P| = 5 * |M_only| = 5 * (|M| - |C ∩ M|).
As both |M| and |C ∩ M| are unknown, the total number of persons (|P|) cannot be uniquely determined. Therefore, Statement-2 alone is insufficient.
Evaluation of Statements 1 and 2 together:
From Statement-1, we have |M| = 50.
From Statement-2, we have |P| = 5 * (|M| - |C ∩ M|).
Substitute |M| = 50 into the expression from Statement-2: |P| = 5 * (50 - |C ∩ M|) (Equation 1)
Now, substitute |M| = 50 into the fundamental equation derived from the question stem: |P| = 120 + 50 - |C ∩ M| |P| = 170 - |C ∩ M| (Equation 2)
Equating Equation 1 and Equation 2 to solve for |C ∩ M|: 5 * (50 - |C ∩ M|) = 170 - |C ∩ M| 250 - 5 * |C ∩ M| = 170 - |C ∩ M| 250 - 170 = 5 * |C ∩ M| - |C ∩ M| 80 = 4 * |C ∩ M| |C ∩ M| = 20
Now that |C ∩ M| is known, substitute it back into either Equation 1 or Equation 2 to find |P|: Using Equation 2: |P| = 170 - 20 = 150.
Thus, the total number of persons who attended the party is 150. Since the question can be answered uniquely by combining both statements, but not by either statement alone, option (C) is the correct answer.
Incorrect Options:
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Option (A): The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
This is incorrect because, as demonstrated in the evaluation, neither Statement-1 alone nor Statement-2 alone provides sufficient information to determine the total number of persons.
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Option (B): The Question can be answered by using either Statement alone.
This is incorrect because both Statement-1 and Statement-2 individually leave an unknown variable (|C ∩ M| or |M| and |C ∩ M|, respectively), preventing a unique solution for the total number of persons.
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Option (D): The Question cannot be answered even by using both the Statements together.
This is incorrect because, as shown in the combined evaluation, using both statements together allows for the determination of |C ∩ M|, which subsequently leads to a unique value for the total number of persons attending the party.