Find the coefficient of `x^4` in `(1 - 4x)^(-3//5)`

Answer

To find the coefficient of \( x^4 \) in the expansion of \( (1 - 4x)^{-\frac{3}{5}} \), we can use the Binomial Theorem for negative exponents.

### Step-by-Step Solution:

1. **Identify the General Form**:
The Binomial Theorem states that:
\[ (1 - x)^{-n} = \sum_{r=0}^{\infty} \binom{n+r-1}{r} x^r \]
For our problem, we have \( n = \frac{3}{5} \) and \( x = 4x \).

2. **Set Up the Expansion**:
We rewrite the expression:
\[ (1 - 4x)^{-\frac{3}{5}} = \sum_{r=0}^{\infty} \binom{\frac{3}{5} + r - 1}{r} (4x)^r \]

3. **Find the Coefficient of \( x^4 \)**:
We need the term where \( r = 4 \):
\[ \text{Coefficient of } x^4 = \binom{\frac{3}{5} + 4 - 1}{4} (4)^4 \]

4. **Calculate the Binomial Coefficient**:
\[ \binom{\frac{3}{5} + 3}{4} = \binom{\frac{3}{5} + \frac{15}{5}}{4} = \binom{\frac{18}{5}}{4} \]
Using the formula for the binomial coefficient:
\[ \binom{a}{b} = \frac{a(a-1)(a-2)(a-3)}{b!} \]
We have:
\[ \binom{\frac{18}{5}}{4} = \frac{\frac{18}{5} \cdot \frac{13}{5} \cdot \frac{8}{5} \cdot \frac{3}{5}}{4!} \]

5. **Calculate \( 4^4 \)**:
\[ 4^4 = 256 \]

6. **Combine the Results**:
Now, we can combine the results:
\[ \text{Coefficient of } x^4 = \frac{\frac{18 \cdot 13 \cdot 8 \cdot 3}{625}}{24} \cdot 256 \]

7. **Simplify**:
\[ = \frac{18 \cdot 13 \cdot 8 \cdot 3 \cdot 256}{625 \cdot 24} \]

8. **Final Calculation**:
Calculate \( 18 \cdot 13 \cdot 8 \cdot 3 \):
\[ 18 \cdot 13 = 234, \quad 234 \cdot 8 = 1872, \quad 1872 \cdot 3 = 5616 \]
Now, calculate:
\[ \frac{5616 \cdot 256}{625 \cdot 24} \]
First, calculate \( 625 \cdot 24 = 15000 \).

Finally, we compute:
\[ \frac{5616 \cdot 256}{15000} \]

This gives us the coefficient of \( x^4 \).

### Final Answer:
The coefficient of \( x^4 \) in the expansion of \( (1 - 4x)^{-\frac{3}{5}} \) is \( \frac{59904}{625} \).