Question

The coefficient of $x^{8}$ in the expansion of $\left(1 + \frac{x^{2}}{2 !} + \frac{x^{4}}{4 !} + \frac{x^{6}}{6 !} + \frac{x^{8}}{8 !}\right)^{2}$ is

• A. $\frac{1}{315}$
• B. $\frac{2}{315}$
• C. $\frac{1}{105}$
• D. $\frac{1}{210}$

Answer 1

Answer: (A)

Coefficient of $x^{8}$ in
$\left(1 + \frac{x^{2}}{2 !} + \frac{x^{4}}{4 !} + \frac{x^{6}}{6 !} + \frac{x^{8}}{8 !}\right)\left(1 + \frac{x^{2}}{2 !} + \frac{x^{4}}{4 !} + \frac{x^{6}}{6 !} + \frac{x^{8}}{8 !}\right)$
$=1\times \frac{1}{8 !}+\frac{1}{2 !}\times \frac{1}{6 !}+\frac{1}{4 !}\times \frac{1}{4 !}+\frac{1}{6 !}\times \frac{1}{2 !}+\frac{1}{8 !}\times 1$
$=\frac{1}{0 ! 8 !}+\frac{1}{2 ! 6 !}+\frac{1}{4 ! 4 !}+\frac{1}{6 ! 2 !}+\frac{1}{8 ! 0 !}$
$=\frac{1}{8 !}\left(\frac{8 !}{0 ! 8 !} + \frac{8 !}{2 ! 6 !} + \frac{8 !}{4 ! 4 !} + \frac{8 !}{6 ! 2 !} + \frac{8 !}{8 ! 0 !}\right)$
$=\frac{1}{8 !}\left(^{8} C_{0} + ^{8} C_{2} + ^{8} C_{4} + ^{8} C_{6} + ^{8} C_{8}\right)$
$=\frac{2^{8 - 1}}{8 !}=\frac{2^{7}}{8 !}=\frac{1}{315}$