We need three digits numbers.

Since $1+2+3+4+5=15$

So, number of possible triplets for multiple of $15$ is $1\times 2\times 2$

so $=4\times \lfloor 3+4\times 3-1\times 2\times \lfloor 2=32$

Alternate:

divisible by $5=$$4\times 3=12$

Divisible by $3$

$$\begin{gathered} \begin{array}{l}12\to 3, 4, 5\to 3!=6\\ 15\to 2, 3, 4\to 3!=6\\ 24\to 1, 3, 5\to 3!=6\\ 42\to 1, 2, 3\to \underline{3!=6}\end{array} \\ 24 \end{gathered}$$

Divisible by $15$

$1,4,5; 4,1,5; 3,4,5; 4,3,5$

Required No. $=24+12-4=32$