SOLUTION: 123 year 11 students chose the courses they wanted at the school prom dinner. 75 chose a starter, 106 chose a main course and 81 chose a dessert. 67 chose a main course and dessert

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Question 1201071: 123 year 11 students chose the courses they wanted at the school prom dinner. 75 chose a starter, 106 chose a main course and 81 chose a dessert. 67 chose a main course and dessert, 64 chose a starter and a main course while 49 chose a starter and dessert. Everybody ordered at least one course. Find the probability that a student chosen at random chose all three courses
Answer by ikleyn(53956) About Me  (Show Source):
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123 ("year-11") students chose the courses they wanted at the school prom dinner.
75 chose a starter, 106 chose a main course and 81 chose a dessert.
67 chose a main course and dessert,
64 chose a starter and a main course
while 49 chose a starter and dessert.
Everybody ordered at least one course.
Find the probability that a student chosen at random chose all three courses
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The number of all students who chose all three cources is

    123 - (75 + 106 + 81 - 67 - 64 -49) = 41


according to the Inclusion-Exclusion principle.


Therefore, the probability that a student chosen at random chose all three courses is

    41%2F123 = 1%2F3.    ANSWER

Solved.

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On inclusion-exclusion principle,  see this Wikipedia article

https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle


To see many other similar  (and different)  solved problems,  see the lessons

    - Counting elements in sub-sets of a given finite set
    - Advanced problems on counting elements in sub-sets of a given finite set
    - Challenging problems on counting elements in subsets of a given finite set
    - Selected problems on counting elements in subsets of a given finite set
    - Inclusion-Exclusion principle problems

in this site.


Happy learning (!)