The problem asks us to find the coefficient of the $y^3$ term in the binomial expansion of $(x - 2y)^4$.

1. Problem Description

The problem asks us to find the coefficient of the $y^3$ term in the binomial expansion of $(x - 2y)^4$.

2. Solution Steps

We can use the binomial theorem to find the coefficient. The binomial theorem states that

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$.

In our case, we have $(x - 2y)^4$. We want to find the term with $y^3$, so we need to find the term where $k = 3$.

Here, $a = x$, $b = -2y$, and $n = 4$.

So we have:

$\binom{4}{3} x^{4-3} (-2y)^3$

$\binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!} = \frac{4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1)(1)} = 4$

Now, we plug in the values:

$4 \cdot x^{1} \cdot (-2y)^3 = 4 \cdot x \cdot (-8y^3) = -32xy^3$

The coefficient of $y^3$ is $-32x$. However, we are looking for the coefficient of the $y^3$ term, meaning we want the value that $y^3$ is multiplied by.

The term with $y^3$ is given by:

$\binom{4}{3}x^{4-3}(-2y)^3 = \binom{4}{3}x^1(-2y)^3 = 4x(-8y^3) = -32xy^3$.

So, the coefficient of $y^3$ in the binomial expansion of $(x - 2y)^4$ is $-32x$. However, the prompt is slightly ambiguous, but given the answer format used in the question, it is most likely asking for the numerical coefficient of the $xy^3$ term. Therefore we consider the term $-32xy^3$ and observe that the coefficient of $xy^3$ is -

3

2.

3. Final Answer

-32