In this note the distribution of the approximation coefficients $\Theta_n$, associated with the regular continued fraction expansion of numbers $x\in [0,1)$, is given under extra conditions on the numerators and denominators of the convergents $p_n/q_n$. Similar results are also obtained for $S$-expansions. Further, a Gauss-Kusmin type theorem is derived for the regular continued fraction expansion under these extra conditions.