Question

The coefficient of $x^{-17}$ in ${(x^4-\frac{1}{x^3})}^{15}$ is:

• A. $1365$
• B. $-1365$
• C. $3003$
• D. $-3003$

Answer 1

The correct option is B $-1365$

Let $(r+1)$th term contains $x^{-17}$
$T_{r+1}={}^{15}{C}_r(x^4{)}^{15-r}{(-\frac{1}{x^3})}^r={}^{15}{C}_r\cdot x^{60-7r}(-1{)}^r$
For $x^{-17},60-7r=-17$
$\Rightarrow r=11$
$\Rightarrow T_{12}={}^{15}{C}_{11}\cdot x^{-17}(-1{)}^{11}$
Hence, coefficient of $x^{-17}$ is $-1365$.