What is the coefficient of $x^2y^6$ in the expansion of $\left(\frac{3}{5}x-\frac{y}{2}\right)^8$?  Express your answer as a common fraction.
By the Binomial theorem, this term is $$\binom82 \left(\frac{3}{5}x\right)^2\left(-\frac{y}{2}\right)^6=28\cdot\frac{9}{25}\cdot\frac{1}{64}x^2y^6=\boxed{\frac{63}{400}}x^2y^6$$
\frac{63}{400}
Level 4
Counting & Probability
Problem: What is the coefficient of $x^2y^6$ in the expansion of $\left(\frac{3}{5}x-\frac{y}{2}\right)^8$?  Express your answer as a common fraction.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the coefficient of \(x^2y^6\) in the expansion of \(\left(\frac{3}{5}x - \frac{y}{2}\right)^8\), we apply the binomial theorem, which states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = \frac{3}{5}x\), \(b = -\frac{y}{2}\), and \(n = 8\).

We need to identify the terms in the expansion that will yield \(x^2y^6\). This requires selecting \(k\) such that the power of \(y\) is 6, which means we need \(n-k = 2\) (the power of \(x\)). Therefore, we set up the equations:

1. \(n - k = 2\) implies \(k = 8 - 2 = 6\).
2. The term corresponding to \(k = 6\) in the expansion is given by \(\binom{8}{6} \left(\frac{3}{5}x\right)^{2} \left(-\frac{y}{2}\right)^{6}\).

Next, we calculate each component:

- The binomial coefficient \(\binom{8}{6} = \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28\).
- The term \(\left(\frac{3}{5}x\right)^{2} = \left(\frac{3}{5}\right)^{2} x^{2} = \frac{9}{25} x^{2}\).
- The term \(\left(-\frac{y}{2}\right)^{6} = \left(-\frac{1}{2}\right)^{6} y^{6} = \frac{1}{64} y^{6}\).

Now, we combine these components to find the coefficient of the term:

\[
\text{Coefficient} = \binom{8}{6} \cdot \left(\frac{3}{5}\right)^{2} \cdot \left(-\frac{1}{2}\right)^{6} = 28 \cdot \frac{9}{25} \cdot \frac{1}{64}.
\]

Calculating this step-by-step:

1. First, calculate \(28 \cdot \frac{9}{25} = \frac{252}{25}\).
2. Next, multiply by \(\frac{1}{64}\):

\[
\frac{252}{25} \cdot \frac{1}{64} = \frac{252}{1600}.
\]

Now, simplify \(\frac{252}{1600}\). The greatest common divisor of 252 and 1600 is 4:

\[
\frac{252 \div 4}{1600 \div 4} = \frac{63}{400}.
\]

Thus, the final answer is \(\frac{63}{400}\).<|END_OF_LATENT|> The final answer is $\frac{63}{400}$.