Question

The value of $(2.{}^1{P}_0-3.{}^2{P}_1+4.{}^3{P}_2-\dots .$ up to$51^{\text{th }}$ term $)+(1!-2!+3!-\dots \dots$ up to $51^{\text{th }}$ term $)$ is equal to :

• A. $1+(51)!$
• B. $1-51(51)!$
• C. $1+(52)!$
• D. 1

Answer 1

Answer: (C)

$S=(2\cdot {}^1{p}_0-3\cdot {}^2{p}_1+4\cdot {}^3{p}_2\dots \dots \dots$ upto 51 terms $)$
$+(1!+2!+3!\dots \dots \dots$ upto 51 terms $)$
$\left[\because {}^n{p}_{n-1}=n!\right]$
$\therefore S=(2\times 1!-3\times 2!+4\times 3!\dots +52.51!)$
$+(1!-2!+3!\dots \dots \dots (51)!)$
$=(2!-3!+4!\dots \dots +52!)$
$+(1!-2!+3!-4!+\dots \dots +(51)!)$
$=1!+52!$