The sum of rational coefficients in expansion of ${(\sqrt[3]{5}x+\sqrt{3}y+z)}^6$ is

Any term of the expansion is of the form $\frac{6!}{a!b!c!}{\left(5^{1/3}x\right)}^a{\left(3^{1/2}y\right)}^b{z}^c$

$z^c,a,b,c$ non-negative integers and $a+ b+ c= 6.$ For rational coefficients ‘a’ must be multiple of 3 and b must be multiple of 2.  The following are the possibilities 

Sum of coefficients $=\frac{6!}{6!}+\frac{6!}{2!4!}.3+\frac{6!}{4!2!}{3}^2+\frac{6!}{6!}{3}^3+\frac{6!}{3!3!}5+\frac{6!}{3!2!}\mathrm{.5.3}+\frac{6!}{6!}{5}^2=1233$