Question:
The value of $\left(2 \cdot{ }^{1} P_{0}-3 \cdot{ }^{2} P_{1}+4 \cdot{ }^{3} P_{2}-\ldots\right.$ up to $51^{\text {th }}$ term $)$
$+\left(1 !-2 !+3 !-\ldots\right.$ up to $51^{\text {th }}$ term $)$ is equal to :
Correct Option: , 3
Solution:
We know, $(r+1) \cdot{ }^{r} P_{r-1}=(r+1) \cdot \frac{r !}{1 !}=(r+1) !$
So, $\left(2 \cdot{ }^{1} P_{0}-3 \cdot{ }^{2} P_{1}+\ldots . .51\right.$ terms $)+$
$(1 !-2 !+3 !-\ldots$ upto 51 terms)
$=[2 !-3 !+4 !-\ldots+52 !]+[1 !-2 !+3 !-\ldots+51 !]$
$=52 !+1 !=52 !+1$
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