Find the fourth term in the expansion of `(x - 2y)^12`

Answer

To find the fourth term in the expansion of \((x - 2y)^{12}\), we can use the Binomial Theorem. The Binomial Theorem states that:

\[ (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]

where \(\binom{n}{r}\) is the binomial coefficient, which can be calculated as:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

In our case, we have \(a = x\), \(b = -2y\), and \(n = 12\). We want to find the fourth term in the expansion, which corresponds to \(r = 3\) (since the term number \(T_{r+1}\) corresponds to \(r\)).

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term \(T_{r+1}\) in the expansion is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]

2. **Substitute Values**:
For the fourth term (\(T_4\)), we have \(r = 3\), \(n = 12\), \(a = x\), and \(b = -2y\). Thus:
\[ T_4 = \binom{12}{3} x^{12-3} (-2y)^3 \]

3. **Calculate the Binomial Coefficient**:
\[ \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = \frac{1320}{6} = 220 \]

4. **Calculate the Powers**:
\[ x^{12-3} = x^9 \]
\[ (-2y)^3 = -8y^3 \]

5. **Combine the Results**:
Now substitute back into the term:
\[ T_4 = 220 \cdot x^9 \cdot (-8y^3) = 220 \cdot (-8) \cdot x^9 \cdot y^3 = -1760 x^9 y^3 \]

### Final Answer:
The fourth term in the expansion of \((x - 2y)^{12}\) is:
\[ \boxed{-1760 x^9 y^3} \]