Question

Find the coefficient of $x^6$ in the expansion of ${(a{x}^2+bx+c)}^9$

• A. $84b^6{C}^3+630a{b}^4{c}^4+75a^2{b}^2{c}^5+84a^3{c}^6$
• B. $72b^6{C}^3+540a{b}^4{c}^4+75a^2{b}^2{c}^5+72a^3{c}^6$
• C. $84b^6{C}^3+540a{b}^4{c}^4+80a^2{b}^2{c}^5+84a^3{c}^6$
• D. $72b^6{C}^3+630a{b}^4{c}^4+80a^2{b}^2{c}^5+72a^3{c}^6$

Answer 1

Answer: (A)

$84b^6{C}^3+630a{b}^4{c}^4+75a^2{b}^2{c}^5+84a^3{c}^6$

$x^6$ can be obtained in the following ways
$(a{x}^2{)}^3.(bx{)}^0{c}^6$
Coefficient will be =$\frac{9!}{3!.6!}.a^3{c}^6$
$=84a^3{c}^6$
$(a{x}^2{)}^2.(bx{)}^2{c}^5$
Coefficient will be =$\frac{9!}{2!.2!.5!}.a^2.b^2.c^5$
$=756a^2{b}^2{c}^5$
$(a{x}^2{)}^1.(bx{)}^4{c}^4$
Coefficient will be =$\frac{9!}{1!.4!.4!}.a^1.b^4.c^4$
$=630a^1{b}^4{c}^4$
$(a{x}^2{)}^0.(bx{)}^6{c}^3$
Coefficient will be =$\frac{9!}{6!.3!}.a^0.b^6.c^3$
$=84a^0{b}^6{c}^3$