How many numbers between $400$ and $600$ begin with or end with the digit $5$?

We need to find the number of integers strictly between $400$ and $600$ (i.e., from $401$ to $599$) that begin with $5$ OR end with $5$.
Let $A$ be the set of numbers that begin with $5$.
Let $B$ be the set of numbers that end with $5$.

1. **Numbers beginning with $5$ ($500$ to $599$):**
These are all the numbers from $500$ to $599$.

$$n(A)=100$$

2. **Numbers ending with $5$ between $400$ and $600$:**
These are numbers of the form $\text{\_ \_ 5}$.
* For the hundreds digit, it can be either $4$ or $5$.
* For the tens digit, it can be any digit from $0$ to $9$ ($10$ choices).

Thus, $n(B)=2\times 10=20$.
3. **Numbers that both begin with $5$ AND end with $5$ ($A\cap B$):**
These are numbers of the form $5\text{\_ }5$.
* The tens digit can be any digit from $0$ to $9$ ($10$ choices).

Thus, $n(A\cap B)=10$.

Using the principle of inclusion-exclusion, the total count of numbers that begin with or end with $5$ is:

$$\text{Total}=n(A)+n(B)-n(A\cap B)=100+20-10=110$$