Find the coefficient of `x^256` in `(1-x)^101*(x^2+x+1)^100`

Options

A. `"^100C_15`
B. `- "^100C_15`
C. `"^100C_18`
D. `"^100C_16`

Answer

To find the coefficient of \( x^{256} \) in the expression \( (1-x)^{101} (x^2 + x + 1)^{100} \), we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the expression:
\[ (1-x)^{101} (x^2 + x + 1)^{100} \]
We can express \( (x^2 + x + 1)^{100} \) in a more manageable form. Notice that \( x^2 + x + 1 = \frac{1 - x^3}{1 - x} \). Thus, we can rewrite:
\[ (x^2 + x + 1)^{100} = \left(\frac{1 - x^3}{1 - x}\right)^{100} = (1 - x^3)^{100} (1 - x)^{-100} \]

### Step 2: Combine the expressions
Now, substituting back into our original expression, we have:
\[ (1-x)^{101} (1 - x^3)^{100} (1 - x)^{-100} \]
This simplifies to:
\[ (1-x)^{1} (1 - x^3)^{100} = (1 - x)(1 - x^3)^{100} \]

### Step 3: Expand the expression
Next, we need to expand \( (1 - x)(1 - x^3)^{100} \). The first part, \( (1 - x) \), contributes a factor of \( -x \) when we consider the coefficient of \( x^{256} \). Thus, we need to find the coefficient of \( x^{255} \) in \( (1 - x^3)^{100} \).

### Step 4: Find the coefficient of \( x^{255} \) in \( (1 - x^3)^{100} \)
Using the binomial theorem, we can expand \( (1 - x^3)^{100} \):
\[ (1 - x^3)^{100} = \sum_{k=0}^{100} \binom{100}{k} (-1)^k (x^3)^k = \sum_{k=0}^{100} \binom{100}{k} (-1)^k x^{3k} \]
We need the coefficient of \( x^{255} \), which means we need \( 3k = 255 \). Solving for \( k \):
\[ k = \frac{255}{3} = 85 \]

### Step 5: Calculate the coefficient
Now, we find the coefficient corresponding to \( k = 85 \):
\[ \text{Coefficient} = \binom{100}{85} (-1)^{85} \]
Since \( (-1)^{85} = -1 \), we have:
\[ \text{Coefficient} = -\binom{100}{85} \]

### Step 6: Final result
Since we are looking for the coefficient of \( x^{256} \) in the original expression, we have:
\[ \text{Coefficient of } x^{256} = -\binom{100}{85} \]
Using the property of binomial coefficients, we know:
\[ \binom{100}{85} = \binom{100}{15} \]
Thus, the final answer is:
\[ \text{Coefficient of } x^{256} = \binom{100}{15} \]

### Conclusion
The coefficient of \( x^{256} \) in \( (1-x)^{101} (x^2 + x + 1)^{100} \) is:
\[ \boxed{\binom{100}{15}} \]