I need to evaluate $$\fbox{$\dfrac{\sum_{k=0}^r{{n \choose k } \cdot {{n-2k} \choose {r-k}}}}{\sum_{k=r}^{n}{{n \choose k} \cdot {2k \choose 2r} \cdot \left(\frac{3}{4}\right)^{n-k} \cdot \left(\frac{1}{2}\right)^{2k-2r}}}$}$$
$\underline{\text{My attempt}}$ $\require{cancel}$
I expanded the binomial coefficients in denominator as $$\dfrac{n! \cdot (2k)!}{k!(n-k)! \cdot (2r)!(2k-2r)!}$$ and took 2 outside the factorial terms which resulted in $$\dfrac{n! \cdot \cancel{2^{k}} \cdot k!}{k!(n-k)! \cdot \cancel{2^{r}} \cdot r! \cdot \cancel{2^{k-r}} \cdot (k-r)!}$$ How to proceed further?
Edit: I realised the error in my approach. Please ignore it and suggest another approach.