If $\sum _{r=0}^{25} \left\{{ }^{50}{C}_r 50-r{C}_{25-r}\right\}=k\left({ }^{50}{C}_{25}\right)$ then k, is equal to

we have

$\begin{array}{l}\sum _{r=0}^{25}{ }^{50}{C_r}^{50\cdots r}{C}_{25-r}=k\left({ }^{50}{C}_{25}\right)\\ \Rightarrow \sum _{r=0}^{25} \frac{50!}{(50-r)!r!}\times \frac{(50-r)!}{25!(25-r)!}=k\cdot \frac{50!}{(25!{)}^2}\\ \Rightarrow \sum _{r=0}^{25} \frac{25!}{r!(25-r)!}=k\\ \Rightarrow \sum _{r=0}^{25}{ }^{25}{C}_r=k\Rightarrow k=2^{25}\end{array}$