A) \[^{2n}{{C}_{n}}\]
B) \[^{2n}{{C}_{n-1}}\]
C) \[^{2n}{{C}_{n-2}}\]
D) None of these
Correct Answer: A
Solution :
\[\frac{\text{The}\,\,\text{coefficient}\,\,\text{of}\,\,{{t}_{r+1}}}{\text{The}\,\,\text{coefficient}\,\,\text{of}\,\,{{t}_{r}}}=\frac{^{2n}{{C}_{r}}}{^{2n}{{C}_{r-1}}}\] \[=\frac{2n-r+1}{r}\] The coefficient of\[{{t}_{r+1}}\ge \]The coefficient of\[{{t}_{r}}\], provided \[\frac{2n-r+1}{r}\ge 1\]or\[2n+1\ge 2r\] or \[r\le \frac{2n+1}{2}\]or\[r\le n+\frac{1}{2}\] Hence, the greatest coefficient = The coefficient of\[~(n+\text{1})\text{th}\]term \[{{=}^{2n}}{{C}_{n}}=\frac{(2n)!}{{{(n!)}^{2}}}\]
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