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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 appreciated.

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