On an international flight carrying 150 passengers, each passenger was offered three meal … — JAMB UTME

On an international flight carrying $150$ passengers, each passenger was offered three meal options: Chicken ( $C$ ), Fish ( $F$ ), and Vegetarian ( $V$ ). A survey of their choices showed that $75$ passengers chose Chicken, $60$ chose Fish, and $50$ chose Vegetarian. Additionally, $15$ passengers chose both Chicken and Fish, $12$ chose both Fish and Vegetarian, $18$ chose both Chicken and Vegetarian, while $8$ passengers chose all three meals. How many passengers chose none of the three meal options?

Answer: 2 passengers

Answer

2 passengers chose none of the meal options.

Using the inclusion-exclusion principle for three overlapping sets, the total number of passengers taking at least one meal is calculated as

|C \cup F \cup V| = 75 + 60 + 50 - (15 + 12 + 18) + 8 = 148

. Subtracting this value from the total count of

150

passengers yields

150 - 148 = 2

passengers who selected none of the meal options.

Step-by-Step Solution

Identify the cardinalities of the individual sets, pairwise intersections, triple intersection, and the universal set.

N(U) = 150

|C| = 75

|F| = 60

|V| = 50

|C \cap F| = 15

|F \cap V| = 12

|C \cap V| = 18

, and

|C \cap F \cap V| = 8

Organizing the given information allows for direct application of set cardinality formulas.

Calculate the total number of passengers who selected at least one meal using the Principle of Inclusion-Exclusion for three sets.

|C \cup F \cup V| = 75 + 60 + 50 - (15 + 12 + 18) + 8 = 185 - 45 + 8 = 148

Adding individual set totals overcounts elements in pairwise intersections, and subtracting pairwise intersections subtracts the triple intersection one too many times, so it must be added back.

Find the number of passengers who selected none of the meal choices by taking the complement of the union with respect to the universal set.

N(\text{None}) = N(U) - |C \cup F \cup V| = 150 - 148 = 2

Passengers choosing none of the meal options correspond to the region outside all three sets within the universal set.

Key Concept

Principle of Inclusion-Exclusion for Three Sets and Complement of Union

Estimated Time:1m 30s

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