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

1(51)!(51)!1!50!+(51)!3!48!+…..+(51)!51!0!1(51)!51C1+51C3+…..+51C51=1(51)!2512=250(51)!

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

1(51)!(51)!1!50!+(51)!3!48!+…..+(51)!51!0!1(51)!51C1+51C3+…..+51C51=1(51)!2512=250(51)!

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The value of $\frac{1}{1! 50!} + \frac{1}{3! 48!} + \frac{1}{5! 46!} + \dots . + \frac{1}{49! 2!} + \frac{1}{51! 1!}$ is :

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$\frac{2^{51}}{50!}$

$\frac{2^{50}}{50!}$

$\frac{2^{51}}{51!}$

$\frac{2^{50}}{51!}$

answer is D.

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Detailed Solution

$\begin{array}{l} \frac{1}{(51)!} [\frac{(51)!}{1! 50!} + \frac{(51)!}{3! 48!} + \dots . . + \frac{(51)!}{51! 0!}] \\ \frac{1}{(51)!} [51 C_{1} + 51 C_{3} + \dots . . + 51 C_{51}] \\ = \frac{1}{(51)!} [\frac{2^{51}}{2}] = \frac{2^{50}}{(51)!} \end{array}$