A biotechnology consortium surveyed 180 research laboratories regarding their implementati… — GRE General Test

A biotechnology consortium surveyed $180$ research laboratories regarding their implementation of three diagnostic platforms: Platform $A$ , Platform $B$ , and Platform $C$ .

- $95$ laboratories use Platform $A$ .
- $85$ laboratories use Platform $B$ .
- $80$ laboratories use Platform $C$ .
- $15$ laboratories use all three platforms.
- $20$ laboratories use none of the three platforms.
- The number of laboratories that use Platform $A$ and Platform $B$ but NOT Platform $C$ is equal to the number of laboratories that use Platform $B$ and Platform $C$ but NOT Platform $A$ .
- $30$ laboratories use Platform $A$ and Platform $C$ but NOT Platform $B$ .

How many laboratories use Platform $A$ ONLY?

A
15
B
20
30Answer
D
45
E
50

Answer

30 laboratories use Platform A only.

Subtracting the 20 laboratories that use none of the platforms from the total population of 180 gives a union size of 160. Applying the inclusion-exclusion formula yields

160 = (95 + 85 + 80) - (x + 15 + x + 15 + 30 + 15) + 15

, which resolves to

x = 20

laboratories that use Platforms A and B only. Subtracting all overlap regions from Platform A's total gives

95 - 20 - 30 - 15 = 30

laboratories that use Platform A only.

Step-by-Step Solution

Calculate the total number of laboratories that use at least one platform.

|A \cup B \cup C| = 180 - 20 = 160

Laboratories using at least one platform equal the total surveyed minus those using none.

Set up the Principle of Inclusion-Exclusion for three sets.

|A \cup B \cup C| = |A| + |B| + |C| - (|A \cap B| + |B \cap C| + |A \cap C|) + |A \cap B \cap C|

Standard formula relating total union size to individual set sizes and intersections.

Express pairwise intersections in terms of non-overlapping regions and solve for the unknown region

x

Let

x

be the number of labs using Platform

A

and

B

only. Given

A \cap C \text{ only} = 30

and

A \cap B \cap C = 15

, we have:

160 = (95 + 85 + 80) - [(x + 15) + (x + 15) + (30 + 15)] + 15

160 = 260 - (2x + 75) + 15 = 200 - 2x \implies 2x = 40 \implies x = 20

Substituting the given equality of regions allows finding

x = 20

Calculate the number of laboratories that use Platform

A

only.

\text{Platform } A \text{ only} = |A| - (A \cap B \text{ only}) - (A \cap C \text{ only}) - (A \cap B \cap C) = 95 - 20 - 30 - 15 = 30

Subtracting all overlap regions containing Platform A from the total Platform A count gives the single-region count.

Key Concept

Three-Set Principle of Inclusion-Exclusion and Venn Diagram Region Partitioning

Question

Answer

Step-by-Step Solution

Key Concept

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