The value of 11!,50!+13!,48!+15!,46!++149!,2!+151!,1! is

Question

The value of $\dfrac{1}{1!\,50!}+\dfrac{1}{3!\,48!}+\dfrac{1}{5!\,46!}+\cdots+\dfrac{1}{49!\,2!}+\dfrac{1}{51!\,1!}$ is

Accepted Answer

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

Solutions · 1