$$ \begin{aligned} & \mathrm{S}=\frac{1}{1 ! 50 !}+\frac{1}{3 ! 48 !}+\frac{1}{5 ! 46 !}+\ldots \ldots+\frac{1}{49 ! 2 !}+\frac{1}{51 ! 1 !} \\\\ & =\frac{1}{51 !}\left(\frac{51 !}{1 ! 50 !}+\frac{51 !}{3 ! 48 !}+\frac{51 !}{5 ! 46 !}+\ldots . .+\frac{51 !}{49 ! 2 !}+\frac{51 !}{51 ! 0 !}\right) \\\\ & =\frac{1}{51 !}\left({ }^{51} C_{50}+{ }^{51} C_{48}+{ }^{51} C_{46}+\ldots \ldots .+{ }^{51} C_2+{ }^{51} C_0\right) \\\\ & \because{ }^n C_0+{ }^n C_2+{ }^n C_4+\ldots \ldots=2^{n-1} \\\\ & \therefore S=\frac{2^{50}}{51 !} \end{aligned} $$