A) \[^{2n}{{P}_{n}}\]
B) \[^{2n}{{C}_{n}}\]
C) \[(2n)!\]
D) None of these
Correct Answer: B
Solution :
[b] \[\underset{x\to 1}{\mathop{\lim }}\,\frac{(1-x)(1-{{x}^{2}})...(1-{{x}^{2n}})}{{{\{(1-x)(1-{{x}^{2}})...(1-{{x}^{n}})\}}^{2}}}\] \[=\underset{x\to 1}{\mathop{\lim }}\,\frac{\left( \frac{1-x}{1-x} \right)\left( \frac{1-{{x}^{2}}}{1-x} \right)...\left( \frac{1-{{x}^{2n}}}{1-x} \right)}{{{\left( \left( \frac{1-x}{1-x} \right)\left( \frac{1-{{x}^{2}}}{1-x} \right)...\left( \frac{1-{{x}^{n}}}{1-x} \right) \right)}^{2}}}\] \[=\frac{1\times 2\times 3...(2n)}{{{(1\times 2\times 3...n)}^{2}}}=\frac{(2n)!}{n!n!}{{=}^{2n}}{{C}_{n}}\]
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