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

$\because (r+1){\cdot }^r{P}_{r-1}=(r+1)\cdot \frac{\mid r}{L1}=\underset{\_}{\mid r+1}$

 $\left(2{\cdot }^1{\mathrm{P}}_0-3{\cdot }^2{\mathrm{P}}_1+4{\cdot }^3{\mathrm{P}}_2-\dots \right.$up $\text{ to }\left.{51}^{\text{th }}\text{ term }\right)+\left(1!-2!+3!-\dots \text{ up to }{51}^{\text{th }}\right.$term)

$=[\lfloor 2-\lfloor 3+\lfloor 4-\dots \dots +\lfloor 52]+[\lfloor 1-\lfloor 2+\lfloor 3-\dots \dots +\lfloor 51]$

$=52+\lfloor 1=\lfloor 52+1$