To help plan the number of meals to be prepared in a college cafeteria, a survey was conducted, and the following data were obtained. 125 students ate breakfast. 177 students ate lunch. 270 students ate dinner. 62 students ate breakfast and lunch. 113 students ate breakfast and dinner. 94 students ate lunch and dinner. 54 students ate all three meals. How many students ate (a) At least one meal in the cafeteria? students (b) Exactly one meal in the cafeteria? students (c) Only dinner in the cafeteria? students (d) Exactly two meals in the cafeteria? students
To help plan the number of meals to be prepared in a college cafeteria, a survey was conducted, and the following data were obtained.
177 students ate lunch.
270 students ate dinner.
62 students ate breakfast and lunch.
113 students ate breakfast and dinner.
94 students ate lunch and dinner.
54 students ate all three meals.
How many students ate
students
(b) Exactly one meal in the cafeteria?
students
(c) Only dinner in the cafeteria?
students
(d) Exactly two meals in the cafeteria?
students
Sol:-
We can solve this problem by using the principle of inclusion-exclusion, which states that the number of elements in the union of two or more sets is equal to the sum of the sizes of the individual sets minus the sizes of their intersections, plus the sizes of their pairwise intersections, minus the size of their triple intersections, and so on.
(a) At least one meal in the cafeteria: The total number of students who ate at least one meal in the cafeteria is the union of the three sets: breakfast, lunch, and dinner. Using the principle of inclusion-exclusion, we have:
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