Question

In the expansion of ${(x^3-\frac{1}{x^2})}^n$, $n\in N$, if the sum of the coefficients of $x^5$ and $x^{10}$ is $0$, then $n$ is

• A. $25$
• B. $20$
• C. $15$
• D. $Noneofthses$

Answer 1

Answer: (C)

$15$

For the above sum
$T_{r+1}=(-1{)}^r{}^n{C}_r{x}^{3n-5r}$.
Therefore coefficient of $x^5$ will be
$T_6=-{}^n{C}_5$ ....(i)
And coefficient of $x^{10}$ will be
$T_{11}={}^n{C}_{10}$
Now it is given
${}^n{C}_{10}-{}^n{C}_5=0$
${}^n{C}_{10}={}^n{C}_5$
Therefore
$n-10=5$
$n=15$