A) \[\frac{(2n+1)\,!}{n!(n+1)!}\]
B) \[\frac{(2n+2)!}{n!(n+1)!}\]
C) \[\frac{(2n+1)!}{{{[(n+1)!]}^{2}}}\]
D) \[\frac{(2n)!}{{{(n!)}^{2}}}\]
Correct Answer: A
Solution :
\[\frac{{{T}_{r+1}}}{{{T}_{r}}}=\frac{N-r+1}{r}.x\] Here, N = 2n +1 Þ \[\frac{{{T}_{r+1}}}{{{T}_{r}}}=\frac{2n+2-r}{r}.x\] \[\therefore \] \[{{T}_{r+1}}\ge {{T}_{r}}\] \[\Rightarrow \] \[2n+2-r\ge r\] \[\Rightarrow \] \[2n+2\ge 2r\]Þ \[r\le n+1\] \[\therefore \,\,\,\,\,r=n\] \[{{T}_{r+1}}={{T}_{n+1}}={{\,}^{2n+1}}{{C}_{n+1}}\]\[=\frac{(2n+1)\,!}{(n+1)!\,n!}\].
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