Question

Show that the coefficient of $x^5$ in the expansion of ${(1+x^2)}^5(1+x{)}^4$ is $60$ .

Answer 1

$N=(1+x^2{)}^5(1+x{)}^4$

Coefficient of $x^5$ in N

$=$coefficient of $x^5$ in $(1+x{)}^4(=0)+$ coefficient of $x^3$ in $(1+x{)}^4\times$ coefficient of $x^2$ in $(1+x^2{)}^5+$ coefficient of x in $(1+x{)}^4\times$ coefficient of $x^4$ in $(1+x^2{)}^5$

$(1+x{)}^4=1+4C_1\cdot x+4C_2{x}^2+4C_3{x}^3+4C_4{x}^4$

$=1+4x+6x^2+4x^3+x^4$

$(1+x^2{)}^5=1+5C_1{x}^2+5C_2{x}^4+...\dots +x^{10}$

$=1+5x^2+10x^4+...\dots .+x^{10}$

Coefficient of $x^5=4\times 10+4\times 5$

$=40+20$

$=60$.