If

$\sum_{r=0}^{25}\left({ }^{50} C_{r} \cdot{ }^{50-r} C_{25-r}\right)=\sum_{r=0}^{25}\left(\frac{\lfloor 50}{|50-r| r} \frac{\mid 50-r}{|25| 25-r}\right)$

$=\sum_{r=0}^{25}\left(\frac{\lfloor 50}{\lfloor 25} \times \frac{1}{\lfloor 25} \times\left(\frac{\lfloor 25}{\lfloor 25-r \mid r}\right)\right)$

$={ }^{50} C_{25} \sum_{r=0}^{25}{ }^{25} C_{r}={ }^{50} C_{25}\left(2^{25}\right)$

Then, by comparison, $K=2^{25}$