# The value of

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 :

1. (1) $1-51(51) !$

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

3. (3) $1+(52) !$

4. (4) 1

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)

$=[2 !-3 !+4 !-\ldots+52 !]+[1 !-2 !+3 !-\ldots+51 !]$

$=52 !+1 !=52 !+1$