The value of 1 1 ! 50 ! + 1 3 ! 48 ! + 1 5 ! 46 ! + . . . + 1 49 ! 2 ! + 1 51 ! 1 ! is [2023]

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Solution

Q.
The value of $\frac{1}{1! 50!} + \frac{1}{3! 48!} + \frac{1}{5! 46!} + . . . + \frac{1}{49! 2!} + \frac{1}{51! 1!}$ is [2023]

1 $\frac{2^{51}}{50!}$

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

3 $\frac{2^{51}}{51!}$

4 $\frac{2^{50}}{51!}$

Ans.
(4)

$\frac{1}{1! 50!} + \frac{1}{3! 48!} + \frac{1}{5! 46!} + \dots + \frac{1}{49! 2!} + \frac{1}{51! 1!}$

$= \frac{1}{51!} [\frac{51!}{1! 50!} + \frac{51!}{3! 48!} + \frac{51!}{5! 46!} + \dots + \frac{51!}{49! 2!} + \frac{51!}{51! 1!}]$

$= \frac{1}{51!} [{}^{51}{C_{1}} + {}^{51}{C_{3}} + \dots + {}^{51}{C_{51}}]$

$= \frac{1}{51!} 2^{51 - 1} = \frac{1}{51!} 2^{50}$

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