Question

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

• A. $-83$
• B. $-82$
• C. $-81$
• D. $0$

Answer 1

Answer: (C)

$-81$

$(x^2-x-2{)}^5=(x+1{)}^5.(x-2{)}^5$

$\therefore$ Coefficient of $x^5$ is $=(0,5)+(5,0)+(2,3)+(3,2)+(4,1)+(1,4)$

$=-2^5+1{+}^5{C}_2{.}^5{C}_3(-2{)}^3{+}^5{C}_3{.}^5{C}_2(-2{)}^2{+}^5{C}_4{.}^5{C}_1(-2){+}^5{C}_1{.}^5{C}_4(-2{)}^4$

$=-32+1-800+400-50+400=-81$