Binomial Theorem And Its Simple Applications Question 119
Question: If $ x^{m} $ occurs in the expansion of $ {{(x+1/x^{2})}^{2n}} $ , then the coefficient of $ x^{m} $ is
Options:
A) $ \frac{(2n)!}{(m)!(2n-m)!} $
B) $ \frac{(2n)!3!3!}{(2n-m)!} $
C) $ \frac{(2n)!}{( \frac{2n-m}{3} )!( \frac{4n+m}{3} )!} $
D) none of these
Show Answer
Answer:
Correct Answer: C
Solution:
- [c] $ {T_{r+1}}{{=}^{2n}}C_{r}{x^{2n-r}}{{( \frac{1}{x^{2}} )}^{r}}{{=}^{2n}}C_{r}{x^{2n-3r}} $
This contains $ x^{m},.If 2n-3r=m,then $
$ r=\frac{2n-m}{3} $
$ \Rightarrow $Coefficient\ of\ $x^{m}{{=}^{2n}}C_{r}, $
$ r=\frac{2n-m}{3} $
$ =\frac{2n!}{(2n-r)!r!}=\frac{2n!}{( 2n-\frac{2n-m}{3} )!( \frac{2n-m}{3} )!} $
$ =\frac{2n!}{( \frac{4n+m}{3} )!( \frac{2n-m}{3} )!} $
BETA
