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 :
[c]| \[{{T}_{r+1}}{{=}^{2n}}{{C}_{r}}{{x}^{2n-r}}{{\left( \frac{1}{{{x}^{2}}} \right)}^{r}}{{=}^{2n}}{{C}_{r}}{{x}^{2n-3r}}\] |
| \[This\,\,contains\text{ }{{x}^{m}}\,.If\text{ }2n-3r=m,\,\,then\] |
| \[r=\frac{2n-m}{3}\] |
| \[\Rightarrow Coefficient\,\,of\text{ }{{x}^{m}}{{=}^{2n}}{{C}_{r}},\] |
| \[r=\frac{2n-m}{3}\] |
| \[=\frac{2n!}{(2n-r)!r!}=\frac{2n!}{\left( 2n-\frac{2n-m}{3} \right)!\left( \frac{2n-m}{3} \right)!}\] |
| \[=\frac{2n!}{\left( \frac{4n+m}{3} \right)!\left( \frac{2n-m}{3} \right)!}\] |
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