Re: A farmer sells vegetables to 180 different customers
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28 Jul 2025, 22:55
There are 180 customers in total.
- 90 customers purchased zucchini.
- 115 customers purchased cauliflower.
Let:
- $Z=$ the set of customers who purchased zucchini.
- $C=$ the set of customers who purchased cauliflower.
- $x=$ the number of customers who purchased both zucchini and cauliflower.
Using the principle of inclusion-exclusion:
$$
\(\begin{gathered}
|Z \cup C|=|Z|+|C|-|Z \cap C| \\
|Z \cup C|=90+115-x=205-x
\end{gathered}\)
$$
Since there are 180 customers total, the number purchasing neither zucchini nor cauliflower is:
$$
\(180-|Z \cup C|=180-(205-x)=x-25\)
$$
Quantity A is the number who purchased both zucchini and cauliflower: $x$.
Quantity $B$ is the number who purchased neither: $x-25$.
Since the number of customers cannot be negative, $x$ must be at least 25 to have a valid number for "neither" customers.
Comparing Quantity A and Quantity B:
$$
\(x \quad \text { vs } \quad x-25\)
$$
Clearly,
$$
\(x>x-25\)
$$