Question

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

• A. $\frac{2^{51}}{50 !}$
• B. $\frac{2^{50}}{50 !}$
• C. $\frac{2^{51}}{51 !}$
• D. $\frac{2^{50}}{51 !}$

Answer 1

Answer: (D)

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

Answer 2

$∑_{r=1}^{26} ​\frac{1}{(2r−1)!(51−(2r−1))!} ​$

$= ∑_{r=1}^{26} ​^{51}C_{(2r−1)} \frac{1}{51!}$

$=\frac{1}{51!} {^{51}C_1 + ^{51}C_3 ​+….+ ^{51}C_{51}}= \frac{1}{51!} (2^{50})$

$= (\frac{2^{50}}{51!} )$

Therefore, The correct answer is option (D): $\frac{2^{50}}{51!}$