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

Answer 1

The Correct Option is B

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