In a survey of $400$ students in a school, $100$ were listed as taking apple juice, $150$ as taking orange juice and $75$ were listed as taking both apple as well as orange juice. Find how many students were taking neither apple juice nor orange juice.
In a survey of $400$ students in a school, $100$ were listed as taking apple juice, $150$ as taking orange juice and $75$ were listed as taking both apple as well as orange juice. Find how many students were taking neither apple juice nor orange juice.
Let $U$ denote the set of surveyed students and $A$ denote the set of students taking apple juice and $B$ denote the set of students taking orange juice. Then
$n(U) = 400,n(A) = 100,n(B) = 150$ and $n(A \cap B) = 75$
Now $n\left( {{A^\prime } \cap {B^\prime }} \right) = n{(A \cup B)^\prime }$
${ = n(U) - n(A \cup B)}$
${ = n(U) - n(A) - n(B) + n(A \cap B)}$
${ = 400 - 100 - 150 + 75 = 225\,}$
Hence $225$ students were taking neither apple juice nor orange juice.
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- [KVPY 2017]