Question

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

The correct answer is (B): $^{12}C_{6} + 2$.
Now , $\left(1+ x^{2}\right)^{12} \left(1 +x^{12} + x^{24} + x^{36}\right)$
$= \left[1+ ^{12}C_{1}\left(x^{2}\right) + ^{12}C_{2}\left(x^{2}\right)^{3} + ^{12}C_{3}\left(x^{2}\right)^{3} + ^{12}C_{4}\left(x^{2}\right)^{4} + ^{12}\left(x^{2}\right)_{5} + ^{12}C_{6} \left(x^{2}\right)^{6} +....+ ^{12}C_{12} \left(x^{2}\right)^{12} \right] \times (1 + x^{12} + x^{24} + x^{36})$
Coefficient of $x^{24} = ^{12}C_{6} + ^{12}C_{12} + 1$
$=^{12}C_{6} + 2$