I would like to approximate the following sum when $n \rightarrow \infty$ and $n \gg k$,

$\sum_{x = k}^n \sum_{y > x}^n \frac{\sum_{m = 0}^{k - 1} {y - 2 \choose m}}{\sum_{m = 0}^k {y - 1 \choose m}}$.

The catch is that there is no closed form for partial sum of binomial coefficients. I wonder if there is a good way to approximate $\frac{\sum_{m = 0}^{k - 1} {y - 2 \choose m}}{\sum_{m = 0}^k {y - 1 \choose m}}$ so that the approximation converges to a value given that $n \rightarrow \infty$ and $n \gg k$. Any help is greatly appreciate.

Revision:

I re-thought about my original problem, I believe solving the following problem would make more sense. However it is still unclear to me how to approximate the formula.

Summing ratio of ratio of partial sums of binomial coefficients