What is the coefficient of $x^8$ in the expansion of $(x-1)^9$?
By the Binomial Theorem applied to $(x+(-1))^9$, this term is $\binom98x^8(-1)^1=-9x^8$.  The coefficient of this term is $\boxed{-9}$.
-9
Level 3
Counting & Probability
Problem: What is the coefficient of $x^8$ in the expansion of $(x-1)^9$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the coefficient of \(x^8\) in the expansion of \((x-1)^9\), we can apply the Binomial Theorem. The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this case, we set \(a = x\), \(b = -1\), and \(n = 9\).

Using the theorem, we expand \((x - 1)^9\) as follows:

\[
(x - 1)^9 = \sum_{k=0}^{9} \binom{9}{k} x^{9-k} (-1)^k
\]

We need to identify the term where the power of \(x\) is 8. This occurs when \(9 - k = 8\), which simplifies to \(k = 1\).

Now, we substitute \(k = 1\) into the binomial expansion:

\[
\text{Term for } k = 1: \binom{9}{1} x^{9-1} (-1)^1 = \binom{9}{1} x^8 (-1) = 9 x^8 (-1) = -9 x^8
\]

The coefficient of \(x^8\) in this term is \(-9\). Therefore, the final answer is \(-9\).<|END_OF_LATENT|> The final answer is $-9$.