I think that exponential generating functions can be used for this question. If there must be at least one consonant and at least one vowel , then there can be at most $5$ consonant or vowel.
E.G.F for consonant : $$\bigg(21x+21^2\frac{x^2}{2!}+21^3\frac{x^3}{3!}+21^4\frac{x^4}{4!}+21^5\frac{x^5}{5!}\bigg)$$
E.G.F for vowels : $$\bigg(5x+5^2\frac{x^2}{2!}+5^3\frac{x^3}{3!}+5^4\frac{x^4}{4!}+5^5\frac{x^5}{5!}\bigg)$$
Now find the coefficient of $\frac{x^6}{6!}$ or find $x^6$ and multiply it by $6!$ in the expansion of $$\bigg(21x+21^2\frac{x^2}{2!}+21^3\frac{x^3}{3!}+21^4\frac{x^4}{4!}+21^5\frac{x^5}{5!}\bigg)\bigg(5x+5^2\frac{x^2}{2!}+5^3\frac{x^3}{3!}+5^4\frac{x^4}{4!}+5^5\frac{x^5}{5!}\bigg)$$