Let $(1+\sqrt{2}+\sqrt{3})^{\mathrm{n}}=p_{\mathrm{n}}+q_{\mathrm{n}} \sqrt{2}+r_{\mathrm{n}} \sqrt{3}+s_{\mathrm{n}} \sqrt{6}(n \geqslant 0)$. Find:
a) $\lim _{n \rightarrow \infty} \frac{p_{n}}{q_{n}}$
b) $\lim _{n \rightarrow \infty} \frac{p_{n}}{r_{n}}$
c) $\lim _{n \rightarrow \infty} \frac{p_{n}}{s_{n}}$.