Out of a group of 21 persons, 9 eat vegetables, 10 eat fish and 7 eat eggs. 5 persons eat all three.

dresu9dnjn 2022-05-09 Answered
Out of a group of 21 persons, 9 eat vegetables, 10 eat fish and 7 eat eggs. 5 persons eat all three. How many persons eat at least two out of the three dishes?

My approach: N ( A B C ) = N ( A ) + N ( B ) + N ( C ) N ( A B ) N ( A C ) N ( B C ) + N ( A B C )
21 = 9 + 10 + 7 N ( A B ) N ( A C ) N ( B C ) + 5
N ( A B ) + N ( A C ) + N ( B C ) = 10
Now the LHS has counted N ( A B C ) three times, so I will remove it two times as:-

Number of persons eating at least two dishes = N ( A B + B C + A C ) 2 N ( A B C ) = 10 2 5 = 0
Now it contradicts the questions that there are 5 eating all three dishes.
Is this anything wrong in my approach?
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Answers (1)

heilaritikermx
Answered 2022-05-10 Author has 20 answers
As far as I can see there is something wrong with your approach, but also the question does not have enough information to give a definite answer.

Your mistake is to assume that N ( A B C ) = 21. However presumably the 21 people may include some who eat neither vegetables nor fish nor eggs: if there are n such people then you should have
N ( A B C ) = 21 n   .
If you now follow your method you should get the number of people you want also to be n. And since n is not known, the problem cannot be answered. In fact as you point out, the answer must be at least 5, so we know that n 5. You can find solutions by trial and error with n = 5 , 6 , 7 , 8 , 9 , 10, so any of these is a possible answer.
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