 # 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: $N\left(A\cup B\cup C\right)=N\left(A\right)+N\left(B\right)+N\left(C\right)-N\left(A\cap B\right)-N\left(A\cap C\right)-N\left(B\cap C\right)+N\left(A\cap B\cap C\right)$
$21=9+10+7-N\left(A\cap B\right)-N\left(A\cap C\right)-N\left(B\cap C\right)+5$
$N\left(A\cap B\right)+N\left(A\cap C\right)+N\left(B\cap C\right)=10$
Now the LHS has counted $N\left(A\cap B\cap C\right)$ three times, so I will remove it two times as:-

Number of persons eating at least two dishes $=N\left(A\cap B+B\cap C+A\cap C\right)-2\ast N\left(A\cap B\cap C\right)=10-2\ast 5=0$
Now it contradicts the questions that there are 5 eating all three dishes.
Is this anything wrong in my approach?
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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 $N\left(A\cup B\cup C\right)=21$. 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

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 $n\ge 5$. You can find solutions by trial and error with $n=5,6,7,8,9,10$, so any of these is a possible answer.