How many ways to reach a given tennis-score?

As you saw yourself, you’re looking for the number of permutions of five H’s and three A’s. That’s a string of $8$ symbols, and the A’s can occupy any $3$ of the $8$ positions, so the desired number is simply the number of ways to choose $3$ things from a set of $8$ things. This is the binomial coefficient

$$\binom83=\frac{8!}{3!5!}=\frac{8\cdot7\cdot6}{6}=56\;.$$

You're almost home! You've correctly reduced the problem to "how many ways can we introduce three As into a five Hs". So, of the eight positions which we can label $1$, $2$, $\dots$, $8$, three positions must be As, so a score stream such as HHAAHHAH might be represented as $\{3,4,7\}$.

Hence, we can look at it as "how many $3$-element subsets are there of an $8$-element set".