Key Concepts and Formulas

This problem primarily relies on two fundamental mathematical concepts:

1. Difference of Cubes Formula: The algebraic identity $(a - b)(a^2 + ab + b^2) = a^3 - b^3$. This identity is crucial for simplifying the given expression.
2. Binomial Theorem: For any non-negative integer $n$, the expansion of $(a + b)^n$ is given by the formula: (a + b)^n = \sum_{r=0}^{n} {^n C_r a^{n-r} b^r} = {^n C_0 a^n b^0 + {^n C_1 a^{n-1} b^1 + \dots + {^n C_n a^0 b^n}} where ${^n C_r = \frac{n!}{r!(n-r)!}}$ is the binomial coefficient.
3. Symmetry Property of Binomial Coefficients: ${^n C_r = {^n C_{n-r}}}$. This property helps simplify the final form of the coefficient.

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Step-by-Step Solution

1. Simplify the Given Expression

The given expression is $(1 - x)^{101} (x^2 + x + 1)^{100}$. Our first goal is to simplify this expression using the difference of cubes formula.

• Action: Rewrite $(1 - x)^{101}$ as $(1 - x) (1 - x)^{100}$.
• Reasoning: This allows us to group terms to apply the difference of cubes formula. We aim to combine $(1-x)$ with $(x^2+x+1)$ under the same power. $(1 - x)^{101} (x^2 + x + 1)^{100} = (1 - x) (1 - x)^{100} (x^2 + x + 1)^{100}$
• Action: Combine the terms with the power of 100.
• Reasoning: Using the exponent rule $(a^m b^m = (ab)^m)$, we can group $(1-x)$ and $(x^2+x+1)$. $= (1 - x) [(1 - x)(x^2 + x + 1)]^{100}$
• Action: Apply the difference of cubes formula $(a-b)(a^2+ab+b^2) = a^3-b^3$ with $a=1$ and $b=x$.
• Reasoning: This substitution simplifies the product inside the bracket significantly. Here, $(1 - x)(x^2 + x + 1) = (1)^3 - (x)^3 = 1 - x^3$. Substituting this back into the expression: $= (1 - x) (1 - x^3)^{100}$

2. Expand the Simplified Expression using the Binomial Theorem

Now we need to expand $(1 - x^3)^{100}$ using the binomial theorem.

• Action: Apply the binomial theorem to $(1 - x^3)^{100}$.
• Reasoning: The binomial theorem provides a systematic way to expand expressions of the form $(a+b)^n$. Here, $a=1$, $b=-x^3$, and $n=100$. $(1 - x^3)^{100} = \sum_{r=0}^{100} {^{100} C_r (1)^{100-r} (-x^3)^r}$
• Action: Simplify the terms in the summation.
• Reasoning: $(1)^{100-r}$ is always 1. $(-x^3)^r$ becomes $(-1)^r (x^3)^r = (-1)^r x^{3r}$. $= \sum_{r=0}^{100} {^{100} C_r (-1)^r x^{3r}}$ So, the complete simplified expression is: $(1 - x) \sum_{r=0}^{100} {^{100} C_r (-1)^r x^{3r}}$
• Action: Distribute the $(1-x)$ term into the summation.
• Reasoning: To find the coefficient of a specific power of $x$, we need to separate the terms that will produce powers of $x$ from the first part and the second part of the distributed expression. $= 1 \cdot \left( \sum_{r=0}^{100} {^{100} C_r (-1)^r x^{3r}} \right) - x \cdot \left( \sum_{r=0}^{100} {^{100} C_r (-1)^r x^{3r}} \right)$ $= \sum_{r=0}^{100} {^{100} C_r (-1)^r x^{3r}} - \sum_{r=0}^{100} {^{100} C_r (-1)^r x^{3r+1}}$

3. Identify Terms Contributing to the Coefficient of $x^{256}$

We now need to find which values of $r$ (where $0 \le r \le 100$) will yield an $x^{256}$ term from each of the two summations.

• From the first summation: ${^{100} C_r (-1)^r x^{3r}}$

  • Action: Set the power of $x$ equal to 256.
  • Reasoning: We are looking for the $x^{256}$ term. $3r = 256$
  • Action: Solve for $r$.
  • Reasoning: To determine if a valid term exists. $r = \frac{256}{3}$
  • Explanation: Since $256/3$ is not an integer, there is no term with $x^{256}$ coming from the first summation. The binomial expansion only generates integer powers of $x$.

• From the second summation: $- \sum_{r=0}^{100} {^{100} C_r (-1)^r x^{3r+1}}$

  • Action: Set the power of $x$ equal to 256.
  • Reasoning: We are looking for the $x^{256}$ term. $3r + 1 = 256$
  • Action: Solve for $r$.
  • Reasoning: To determine the specific term. $3r = 255$ $r = \frac{255}{3}$ $r = 85$
  • Explanation: This value of $r=85$ is an integer and falls within the valid range for the summation ($0 \le 85 \le 100$). Therefore, this term contributes to the coefficient of $x^{256}$.

4. Calculate the Coefficient

The coefficient of $x^{256}$ comes only from the second summation when $r=85$. The term from the second summation is: $- {^{100} C_r (-1)^r x^{3r+1}}$ Substitute $r=85$: $- {^{100} C_{85} (-1)^{85}}$

• Action: Evaluate $(-1)^{85}$.
• Reasoning: Any odd power of $(-1)$ results in $-1$. $(-1)^{85} = -1$
• Action: Substitute this value back into the expression for the coefficient. $- {^{100} C_{85} (-1)} = {^{100} C_{85}}$
• Action: Apply the symmetry property of binomial coefficients.
• Reasoning: ${^n C_r = {^n C_{n-r}}}$ is a common simplification and often used to match options in multiple-choice questions. ${^{100} C_{85} = {^{100} C_{100-85}} = {^{100} C_{15}}}$

Thus, the coefficient of $x^{256}$ is ${^{100} C_{15}}$.

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Tips and Common Mistakes

• Don't forget the initial factor: A common mistake is to simplify the $(1-x)^{101}(x^2+x+1)^{100}$ directly to $(1-x^3)^{100}$ without accounting for the extra $(1-x)$ factor. Always ensure all factors are carried through the simplification.
• Integer powers: Remember that the index $r$ in the binomial expansion must be a non-negative integer. If solving for $r$ yields a non-integer, that term does not exist in the expansion.
• Sign management: Be meticulous with signs, especially when dealing with $(-1)^r$. The parity of $r$ determines whether the term is positive or negative.
• Check the range of $r$: Ensure that the calculated value of $r$ is within the bounds of the summation (from 0 to $n$).

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Summary and Key Takeaway

This problem effectively tests your ability to:

1. Recognize and apply fundamental algebraic identities (difference of cubes) for simplification.
2. Apply the binomial theorem correctly to expand polynomial expressions.
3. Systematically identify and combine coefficients of a specific power of $x$ from different parts of an expanded expression.
4. Utilize properties of binomial coefficients for final simplification.

The key takeaway is to simplify complex algebraic expressions as much as possible before attempting to apply expansion theorems. This often involves recognizing common algebraic identities. Also, always account for all factors and ensure the index $r$ is a valid integer within the summation range.