During an urban mobility audit conducted among 400 daily commuters in a metropolitan city,… — State PSC Exam

During an urban mobility audit conducted among $400$ daily commuters in a metropolitan city, data was recorded regarding their regular use of three transit modes: Metro Rail ( $M$ ), Electric Bus ( $E$ ), and Shared Bicycle ( $B$ ). The survey revealed that $190$ commuters use Metro Rail, $160$ use Electric Bus, and $120$ use Shared Bicycle. Furthermore, $60$ commuters use both Metro Rail and Electric Bus, $40$ use both Electric Bus and Shared Bicycle, and $50$ use both Metro Rail and Shared Bicycle. If $20$ commuters utilize all three modes of transport, how many commuters do not use any of these three transit modes?

Answer: 60 commuters

Answer

The number of commuters who do not use any of the three transit modes is 60.

Using the 3-set inclusion-exclusion principle, the total number of commuters using at least one mode is given by

|M \cup E \cup B| = (190 + 160 + 120) - (60 + 40 + 50) + 20 = 340

. Subtracting this from the total sample of 400 yields

400 - 340 = 60

commuters who use none of the three modes.

Step-by-Step Solution

Identify individual set cardinalities and intersections from the problem statement.

Total universe

N = 400

|M| = 190

|E| = 160

|B| = 120

|M \cap E| = 60

|E \cap B| = 40

|M \cap B| = 50

, and

|M \cap E \cap B| = 20

Establishing accurate set values is required before applying set formulas.

Apply the Principle of Inclusion-Exclusion formula for three overlapping sets to find the union

|M \cup E \cup B|

|M \cup E \cup B| = 190 + 160 + 120 - 60 - 40 - 50 + 20 = 340

Pairwise intersections are double-counted when summing individual sets and must be subtracted, while the triple intersection is subtracted thrice and must be added back.

Compute the complement of the union to determine commuters using none of the modes.

Neither mode

= N - |M \cup E \cup B| = 400 - 340 = 60

The complement set represents all members of the universe outside the three-set union.

Key Concept

Principle of Inclusion-Exclusion for Three Sets

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