Question

The value of $\left(2 .{ }^{1} P _{0}-3 .{ }^{2} P _{1}+4 .{ }^{3} P _{2}-\ldots .\right.$ up to$51^{\text {th }}$ term $)+\left(1 !-2 !+3 !-\ldots \ldots\right.$ 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)

$S=\left(2 \cdot{ }^{1} p_{0}-3 \cdot{ }^{2} p_{1}+ 4 \cdot{ }^{3} p_{2} \ldots \ldots \ldots\right.$ upto 51 terms $)$
$+(1 !+2 !+3 ! \ldots \ldots \ldots$ upto 51 terms $)$
$\left[\because{ }^{n} p _{n-1}=n !\right]$
$\therefore S =(2 \times 1 !-3 \times 2 !+4 \times 3 ! \ldots+52.51 !)$
$+(1 !-2 !+3 ! \ldots \ldots \ldots(51) !)$
$=(2 !-3 !+4 ! \ldots \ldots+52 !)$
$+(1 !-2 !+3 !-4 !+\ldots \ldots+(51) !)$
$=1 !+52 !$