Juice problem combinations I have no idea how to start this question... It's impossible in my mind. Help please The owner of a convenience store reports that 500 people who bought bottled fruit juice in a recent week. 400 people bought orange juice 110 bought Apple 50 bought grape 75 bought orange and apple 25 bought orange and grape 30 bought Apple and grape 15 bought orange Apple and grape A.)How many people bought grape and orange but not Apple? B.)How many buyers bought juice other than Apple orange or grape?

Let $O$ be the set of people you bought orange juice.

Let $A$ be the set of people you bought apple juice.

Let $G$ be the set of people you bought grape juice.

Based on the information given, we know the following:

$|O|=400$.

$|A|=110$.

$|G|=50$.

$|O\cap A|=75$.

$|O\cap G|=25$.

$|A\cap G|=30$.

$|O\cap A\cap G|=15$.

For A), the number of people that bought orange and grape but not apple is $|O\cap G|-|O\cap G\cap A|=25-15=10$.

For B), the number of people that bought a juice other than apple, orange or grape is $500-|O\cup A\cup G|$. And, by the inclusion-exclusion formula,

$$ \begin{aligned} |O\cup A\cup G|&=|O|+|A|+|G|-|O\cap A|-|O\cap G|-|A\cap G|+|O\cap A\cap G|\\\ &=400+110+50-75-25-30+15\\\ &=445. \end{aligned} $$

Thus, the answer for B) is $500-445=55$.