Question

The coefficient of $x^5$ in the expansion of $(1+x^2{)}^5(1+x{)}^4$ is

• A. 30
• B. 60
• C. 40
• D. 10
• E. 45

Answer 1

Answer: (B)

We have,
${\left(1+x^2\right)}^5={}^5{C}_0{\left(x^2\right)}^0+{}^5{C}_1{\left(x^2\right)}^1+{}^5{C}_2{\left(x^2\right)}^2$
$+{}^5{C}_3{\left(x^2\right)}^3+{}^5{C}_4{\left(x^2\right)}^4+{}^5{C}_5{\left(x^2\right)}^5$
$=1+5x^2+10x^4+10x^6+5x^8+x^{10}$
$(1+x{)}^4={}^4{C}_0{x}^0+{}^4{C}_1{x}^1+{}^4{C}_2{x}^2+{}^4{C}_3{x}^3+{}^4{C}_4{x}^4$
$=1+4x+6x^2+4x^3+x^4$
$\therefore$ Coefficient of $x^5$ in the product of
${\left(1+x^2\right)}^5(1+x{)}^4$
$=\left(5x^2\right)\cdot \left(4x^3\right)+\left(10x^4\right).(4x)$
$=20x^5+40x^5$
$=60x^5$