A) \[\frac{(2n)!}{(m)!\,(2n-m)!}\]
B) \[\frac{(2n)!\,3!\,3!}{(2n-m)!}\]
C) \[\frac{(2n)!}{\left( \frac{2n-m}{3} \right)\,!\,\left( \frac{4n+m}{3} \right)\,!}\]
D) None of these
Correct Answer: C
Solution :
\[{{T}_{r+1}}={}^{2n}{{C}_{r}}{{x}^{2n-r}}{{\left( \frac{1}{{{x}^{2}}} \right)}^{r}}\] = \[{}^{2n}{{C}_{r}}{{x}^{2n-3r}},\] This contains xm, if 2n - 3r = m i.e. if \[r=\frac{2n-m}{3}\] \ Coefficient of xm \[={}^{2n}{{C}_{r}},\] \[r=\frac{2n-m}{3}\] = \[\frac{2n!}{(2n-r)!r!}=\frac{2n!}{\left( 2n-\frac{2n-m}{3} \right)\,\,!\left( \frac{2n-m}{3} \right)\,\,!}\] = \[\frac{2n!}{\left( \frac{4n+m}{3} \right)\,\,!\,\,\left( \frac{2n-m}{3} \right)\,\,!}\].
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