Number of certain type of words

Question

How to find the number of words consists of two letters $s$ and $t$ of size $k$ that contain exactly three letters $t$ and no adjacent two letters $t$?

Answer 1

The steps can be

$$(\textrm{Fix three t's})\cdot(\textrm{Insert two s' between t})\cdot(\textrm{other works}),$$

and to the other works is distributing indistinguishable items into distinguishable places, so

$$1\cdot1\cdot\binom{(k-3-2)+3}{3},$$

which $(k-3-2)$ is the remaining number of $s$, and with three bars to separate them into four places separated by three $t$'s. E.g.

$$\color{blue}{OO}|||\color{blue}{OO}\to \color{blue}{ss}\ t\ s\ t\ s\ t\ \color{blue}{ss},\\ O|\color{red}{O}||OO\to s\ t\ \color{red}{s}s\ t\ s\ t\ ss.$$