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$stack l i m with x rightwards arrow 1 below fraction numerator left parenthesis 1 minus x right parenthesis left parenthesis 1 minus x to the power of 2 end exponent right parenthesis... left parenthesis 1 minus x to the power of 2 n end exponent right parenthesis over denominator left square bracket left parenthesis 1 minus x right parenthesis left parenthesis 1 minus x to the power of 2 end exponent right parenthesis... left parenthesis 1 minus x to the power of n end exponent right parenthesis right square bracket to the power of 2 end exponent end fraction$ is

(2n) !
$\frac{(2 n)!}{n!}$
$\frac{(2 n)!}{(n!)^{2}}$
none of these

The correct answer is: $fraction numerator left parenthesis 2 n right parenthesis factorial over denominator left parenthesis n factorial right parenthesis to the power of 2 end exponent end fraction$

$stack l i m with x rightwards arrow 1 below fraction numerator left parenthesis 1 minus x right parenthesis left parenthesis 1 minus x to the power of 2 end exponent right parenthesis... left parenthesis 1 minus x to the power of 2 n end exponent right parenthesis over denominator open square brackets left parenthesis 1 minus x right parenthesis left parenthesis 1 minus x to the power of 2 end exponent right parenthesis... left parenthesis 1 minus x to the power of n end exponent right parenthesis close square brackets to the power of 2 end exponent end fraction equals stack l i m with x rightwards arrow 1 below fraction numerator table row cell left parenthesis 1 minus x right parenthesis left parenthesis 1 minus x to the power of 2 end exponent right parenthesis... left parenthesis 1 minus x to the power of n end exponent right parenthesis left parenthesis 1 minus x to the power of n plus 1 end exponent right parenthesis end cell row cell left parenthesis 1 minus x to the power of n plus 2 end exponent right parenthesis.... left parenthesis 1 minus x to the power of 2 n end exponent right parenthesis end cell end table over denominator open square brackets left parenthesis 1 minus x right parenthesis left parenthesis 1 minus x to the power of 2 end exponent right parenthesis... left parenthesis 1 minus x to the power of n end exponent right parenthesis close square brackets to the power of 2 end exponent end fraction$
$equals stack l i m with x rightwards arrow 1 below fraction numerator left parenthesis 1 minus x to the power of n plus 1 end exponent right parenthesis left parenthesis 1 minus x to the power of n plus 2 end exponent right parenthesis... left parenthesis 1 minus x to the power of 2 n end exponent right parenthesis over denominator open square brackets open parentheses 1 minus x close parentheses open parentheses 1 minus x to the power of 2 end exponent close parentheses open parentheses 1 minus x to the power of 3 end exponent close parentheses.... open parentheses 1 minus x to the power of n end exponent close parentheses close square brackets end fraction$
$equals stack l i m with x rightwards arrow 1 below fraction numerator left parenthesis 1 minus x to the power of n plus 1 end exponent right parenthesis left parenthesis 1 minus x to the power of n plus 2 end exponent right parenthesis... left parenthesis 1 minus x to the power of 2 n end exponent right parenthesis over denominator left parenthesis 1 minus x right parenthesis left parenthesis 1 minus x right parenthesis... left parenthesis 1 minus x right parenthesis end fraction cross times fraction numerator left parenthesis 1 minus x right parenthesis left parenthesis 1 minus x right parenthesis.... left parenthesis 1 minus x right parenthesis over denominator left parenthesis 1 minus x right parenthesis left parenthesis 1 minus x right parenthesis to the power of 2 end exponent..... left parenthesis 1 minus x to the power of n end exponent right parenthesis end fraction$
$equals left parenthesis n plus 1 right parenthesis left parenthesis n plus 2 right parenthesis....2 n. fraction numerator 1 over denominator 1 end fraction. fraction numerator 1 over denominator 2 end fraction...... fraction numerator 1 over denominator n end fraction equals fraction numerator left parenthesis 2 n right parenthesis factorial over denominator left parenthesis n factorial right parenthesis to the power of 2 end exponent end fraction$

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