In the expansion of $\left(z^2 + z^{-2}\right)^{15}$, the coefficients of the $8^{th}$ and $19^{th}$ terms are equal. The term independent of $z$ is given by
Step-by-Step Solution
Key Concept: Use the condition that coefficients of $x^8$ and $x^{-8}$ are equal to establish that the corresponding binomial coefficients satisfy $C_r = C_{r'}$.
The general term is $T_{r+1} = C_r (x^8)^{n-r} (x^{-3})^r = C_r x^{8n-11r}$. For the $x^8$ term: $8n - 11r = 8$, giving $r = \frac{8(n-1)}{11}$. For the $x^{-8}$ term: $8n - 11r = -8$, giving $r' = \frac{8(n+1)}{11}$. Since the coefficients are equal: $C_r = C_{r'}$, which means $r + r' = n$. Substituting: $\frac{8(n-1)}{11} + \frac{8(n+1)}{11} = n$, which gives $\frac{16n}{11} = n$, so $n = 25$. Therefore, $^nC_{20} = ^{25}C_{20}$.
Correct Answer: ^{25}C_{20}