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Difficulty: HardOverlapping Sets, Statistics, and Data Distributions

Among a panel of 150150 medical research trials, each trial evaluates at least one of three experimental drugs: Drug XX, Drug YY, or Drug ZZ. If 8080 trials evaluate Drug XX and 7070 trials evaluate Drug YY, how many trials evaluate Drug ZZ only?

(1) Exactly 2525 trials evaluate both Drug XX and Drug YY.
(2) Exactly 4040 trials evaluate at least two of the three drugs, and no trial evaluates all three drugs.

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Answer
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. C
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The correct choice is the option stating that Statement (1) alone is sufficient, but Statement (2) alone is not sufficient. By rephrasing the question, the number of trials evaluating Drug Z only equals the total number of trials (150150) minus the number of trials evaluating Drug X or Drug Y (∣X∪Y∣|X \cup Y|). Statement (1) gives the intersection ∣X∩Y∣=25|X \cap Y| = 25, which immediately yields ∣X∪Y∣=80+70−25=125|X \cup Y| = 80 + 70 - 25 = 125, so Drug Z only =150−125=25= 150 - 125 = 25. Statement (2) only establishes that the double overlap equals 4040, leaving the specific overlap between X and Y unknown.

Step-by-Step Solution

1
Rephrase the target question using set notation.
Let UU be the total set of trials (∣U∣=150|U| = 150). Since every trial evaluates at least one drug, ∣X∪Y∪Z∣=150|X \cup Y \cup Z| = 150. The number of trials evaluating Drug ZZ only is given by ∣Z only∣=∣X∪Y∪Z∣−∣X∪Y∣=150−∣X∪Y∣|Z \text{ only}| = |X \cup Y \cup Z| - |X \cup Y| = 150 - |X \cup Y|.
Simplifying the target target shows that finding ∣X∪Y∣|X \cup Y| is both necessary and sufficient to answer the question.
2
Evaluate Statement (1) independently.
Statement (1) gives ∣X∩Y∣=25|X \cap Y| = 25. Using the standard principle of inclusion-exclusion for two sets: ∣X∪Y∣=∣X∣+∣Y∣−∣X∩Y∣=80+70−25=125|X \cup Y| = |X| + |Y| - |X \cap Y| = 80 + 70 - 25 = 125. Then ∣Z only∣=150−125=25|Z \text{ only}| = 150 - 125 = 25.
Statement (1) yields a single, unique numerical answer (2525), so Statement (1) alone is sufficient.
3
Evaluate Statement (2) independently.
Let a,b,ca, b, c be the double overlaps ∣X∩Y∣|X \cap Y|, ∣Y∩Z∣|Y \cap Z|, and ∣X∩Z∣|X \cap Z|, and d=∣X∩Y∩Z∣=0d = |X \cap Y \cap Z| = 0. Statement (2) states a+b+c=40a + b + c = 40. Total set formula gives 150=80+70+∣Z∣−40  ⟹  ∣Z∣=40150 = 80 + 70 + |Z| - 40 \implies |Z| = 40. However, ∣Z only∣=∣Z∣−(b+c)=40−(40−a)=a=∣X∩Y∣|Z \text{ only}| = |Z| - (b + c) = 40 - (40 - a) = a = |X \cap Y|, which is unknown.
Since the value of ∣X∩Y∣|X \cap Y| can vary, Statement (2) alone does not yield a unique numerical value and is insufficient.

Key Concept

Overlapping Sets and Data Sufficiency Target Rephrasing
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Topics

Overlapping Sets, Statistics, and Data Distributions

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