The number of four-letter words that can be formed with letters $a, b, c$ such that all three letters occur is

Out of given three letters $a, b, c$, we will repeat exactly one of the letter and we can do this in ${}^{3}C_1$ ways. 

Now, we have four letters and out of these four letters two are identical, so number of ways to arrange these four letters is $\frac{4!}{2!}$.

So, the number of four letter words that can be formed with letters $a, b, c$ such that all three letters occur is ${}^{3}C_1\times \frac{4!}{2!}=36$