Solve the Following Questions

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 $)+\left(1 !-2 !+3 !-\ldots . .\right.$ up to $51^{\text {th }}$ term $)$ is equal to :

  1. $1+(51) !$

  2. $1-51(51) !$

  3. $1+(52) !$

  4. 1


Correct Option: , 3

Solution:

$\mathrm{S}=\left(2 \cdot{ }^{1} \mathrm{p}_{0}-3 \cdot{ }^{2} \mathrm{p}_{1}+4 \cdot{ }^{3} \mathrm{p}_{2} \ldots \ldots \ldots . .\right.$ upto 51 terms $)$

$+(1 !+2 !+3 ! \ldots \ldots \ldots .$ upto 51 terms)

$\left[\because{ }^{n} p_{n-1}=n !\right]$

$\therefore \quad S=(2 \times 1 !-3 \times 2 !+4 \times 3 ! \ldots . .+52.51 !)$

$+(1 !-2 !+3 !$................(51)!)

$=(2 !-3 !+4 ! \ldots \ldots \ldots+52 !)$

$+(1 !-2 !+3 !-4 !+\ldots \ldots+(51) !)$

$=1 !+52 !$

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