In a survey of $100$ people, $60$ like tea, $50$ like coffee, and $25$ like both tea and coffee. How many people like neither tea nor coffee?

Let $T$ be the set of people who like tea, and $C$ be the set of people who like coffee.
Given:

$$n(\text{Total})=100$$

$$n(T)=60$$

$$n(C)=50$$

$$n(T\cap C)=25$$

Find the number of people who like at least one beverage using the set union formula:

$$n(T\cup C)=n(T)+n(C)-n(T\cap C)$$

$$n(T\cup C)=60+50-25=85$$

The number of people who like neither beverage is:

$$\text{Neither}=n(\text{Total})-n(T\cup C)=100-85=15$$

Therefore, $15$ people like neither tea nor coffee.