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. None of these

Answer 1

Answer: (B)

$\left(1+x^{2}\right)^{5} \cdot(1+x)^{4}$
$=\left({ }^{5} C_{0}+{ }^{5} C_{1} x^{2}+{ }^{5} C_{2} x^{4}+\ldots\right)$
$\left({ }^{4} C_{0}+{ }^{4} C_{1} x+{ }^{4} C_{2} x^{2}+{ }^{4} C_{3} x^{3}+{ }^{4} C_{4} x^{4}\right)$
The coefficient of $x^{5}$ in $\left[\left(1-x^{2}\right)^{5}(1+x)^{4}\right]$
$={ }^{5} C_{2} \cdot{ }^{4} C_{1}+{ }^{5} C_{1} \cdot{ }^{4} C_{3}$
$=10 \cdot 4+4 \cdot 5$
$=60$