An Erd\H{o}s-Kac theorem for Smooth and Ultra-Smooth integers

We prove an Erd\H{o}s-Kac type of theorem for the set $S(x,y)={n\leq x: p|n \Rightarrow p\leq y }$\mathrm{.}If$\omega (n)$isthenumberofprimefactorsof$n$, we prove that the distribution of $\omega (n)$ for $n\in S(x,y)$ is Gaussian for a certain range of $y$ using method of moments. The advantage of the present approach is that it recovers classical results for the range $u=o(\log \log x )$where$u=\frac{\log x}{\log y}$, with a much simpler proof.