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Let:

$$P_n=\prod_{k=0}^n {n\choose k}$$

...denote the product of all elements in row $n$ of Pascal's Triangle. What is the value of the limit as $n$ goes to $\infty$ of the fraction:

$$\frac{P_{n+1}P_{n-1}}{\left(P_n\right)^2}?$$

$\sqrt{2\pi}$

$2\sqrt{\pi }$

$e$

$\sqrt{2}\,e$

$\sqrt{\pi}\,e$

none of the above