A museum cataloged 250 historical artifacts. Each artifact was subjected to at least one o… — GMAT

A museum cataloged $250$ historical artifacts. Each artifact was subjected to at least one of three preservation procedures: micro-scanning, chemical analysis, or radiocarbon dating. Exactly $140$ artifacts underwent micro-scanning, $120$ underwent chemical analysis, and $100$ underwent radiocarbon dating. If $40$ artifacts underwent both micro-scanning and chemical analysis, $35$ underwent both chemical analysis and radiocarbon dating, and $45$ underwent both micro-scanning and radiocarbon dating, how many artifacts underwent all three procedures?

Answer: 10 artifacts

Answer

10 artifacts underwent all three preservation procedures.

Using the Inclusion-Exclusion Principle for three sets, the total union is equal to the sum of individual set sizes (

140 + 120 + 100 = 360

), minus the sum of two-set overlaps (

40 + 35 + 45 = 120

), plus the three-set overlap. Setting up

250 = 360 - 120 + x

yields

250 = 240 + x

, so

x = 10

Step-by-Step Solution

Set up the Inclusion-Exclusion formula for three overlapping sets.

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

The total size of the union equals the sum of the individual set sizes minus the pairwise intersections plus the triple intersection.

Substitute the given values into the formula.

250 = 140 + 120 + 100 - (40 + 35 + 45) + |A \cap B \cap C|

Since every artifact underwent at least one procedure, the total number of artifacts equals the union of all three sets.

Compute the sum of individual sets and pairwise intersections to isolate the unknown.

250 = 360 - 120 + |A \cap B \cap C| \Rightarrow 250 = 240 + |A \cap B \cap C|

Simplifying the arithmetic terms allows direct calculation of the triple intersection.

Subtract

240

from

250

to find the final value.

|A \cap B \cap C| = 10

Solving the linear equation yields the number of artifacts undergoing all three procedures.

Key Concept

Principle of Inclusion-Exclusion for Three Sets

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic