Question

The coefficient of $x^{24}$ in the expansion of $\left(1+x^{2}\right)^{12}\left(1+x^{12}\right)\left(1+x^{24}\right)$ is

• A. ${ }^{12} C_{6}$
• B. ${ }^{12} C_{6}+2$
• C. ${ }^{12} C_{6}+4$
• D. ${ }^{12} C_{6}+6$

Answer 1

Answer: (B)

${ }^{12} C_{6}+2$

Answer 2

Now, $\left(1+x^{2}\right)^{12}\left(1+x^{12}+x^{24}+x^{36}\right)$
$=\left[1+{ }^{12} C_{1}\left(x^{2}\right)+{ }^{12} C_{2}\left(x^{2}\right)^{2}+{ }^{12} C_{3}\left(x^{2}\right)^{3}\right.$
$+{ }^{12} C_{4}\left(x^{2}\right)^{4}+{ }^{12} C_{5}\left(x^{2}\right)^{5}+{ }^{12} C_{6}\left(x^{2}\right)^{6}$
$\left.+\ldots+{ }^{12} C_{12}\left(x^{2}\right)^{12}\right] \times\left(1+x^{12}+x^{24}+x^{36}\right)$
Coefficient of $x^{24}={ }^{12} C_{6}+{ }^{12} C_{12}+1$
$={ }^{12} C_{6}+2$