Question

The coefficient of $x^5$ in the expansion of $(x^2-x-2{)}^5$ is

• A. $-83$
• B. $-82$
• C. $-86$
• D. $-81$

Answer 1

Answer: (D)

$(x-2{)}^5(x+1{)}^5$
$={[}^5{C}_0{x}^5-{}^5{C}_1{x}^4\times 2+...]{[}^5{C}_0+5C_{1}x+...]$
$\Rightarrow$ Coefficient of $x^5$
$={}^5{C}_0^5{C}_5-{}^5{C}_1\times 2\times {}^5{C}_4+{}^5{C}_2\times 2^2\times {}^5{C}_3-{}^5{C}_3\times 2^3\times {}^5{C}_2$
$+{}^5{C}_4\times 2^4\times {}^5{C}_1-{}^5{C}_5\times 2^5\times {}^5{C}_0$
$=1-5\times 5\times 2+10\times 10\times 4-10\times 10\times 8+5\times 5\times 16-32$
$=-81$