The value of (1/1 ! 50 !)+(1/3 ! 48 !)+(1/5 ! 46 !)+ ldots .+(1/49 ! 2 !)+(1/51 ! 1 !) is :

Question

The value of (1/1 ! 50 !)+(1/3 ! 48 !)+(1/5 ! 46 !)+ ldots .+(1/49 ! 2 !)+(1/51 ! 1 !) is : (A) (251/50 !) (B) (250/50 !) (C) (251/51 !) (D) (250/51 !)

Tardigrade Admin · Accepted Answer

displaystyle∑ r =126 (1/(2 r -1) !(51-(2 r -1)) !)= displaystyle∑ r =126 51 C (2 r -1) (1/51 !) =(1/51 !) 51 C 1+ 51 C 3+ ldots .+ 51 C 51 =(1/51 !)(250)

Q. 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 :

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

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

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

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

$r = 1 \sum 26 \frac{1}{( 2 r - 1 )! ( 51 - ( 2 r - 1 ))!} = r = 1 \sum 26^{51} C_{(2 r - 1)} \frac{1}{51 !}$
$= \frac{1}{51 !} {^{51} C_{1} +^{51} C_{3} + \dots . +^{51} C_{51}} = \frac{1}{51 !} (2^{50})$

Q. The value of 1!50!1​+3!48!1​+5!46!1​+….+49!2!1​+51!1!1​ is :

50!251​

50!250​

51!251​

51!250​