Question

The coefficients of $x^5$ in the expression ${\left(x^2-x-2\right)}^5$ is

• A. -83
• B. -82
• C. -81
• D. -80

Answer 1

Answer: (C)

Coefficients of $x^5=\frac{5!}{a!b!c!}{\left(x^2\right)}^a\cdot (-x{)}^b(-2{)}^c$
$=\frac{5!}{a!b!c!}(-2{)}^c(-1{)}^b{x}^{2a+b}$
Now, $a+b+c=5$ and $2a+b=5$

	a	b	c
i	0	5	0
ii	1	3	1
iii	2	1	2

Coefficient $=\frac{5!}{0!5!0!}(-2{)}^0(-1{)}^5+\frac{5!}{1!3!1!}(-2{)}^1(-1{)}^3+\frac{5!}{2!1!2!}(-2{)}^2(-1{)}^1$
$=-1+40-120=-81$