Question

The coefficient of \( x^{50} \) in \( (1 + x)^{101} (1 - x + x^2)^{100} \) is:

• A. 1
• B. -1
• C. 0
• D. 2

Hint:

When finding the coefficient of a term in an expansion involving binomials, check the exponents to ensure they match required multiples where necessary.

Answer 1

The Correct Option is C

The expression is: \[ (1 + x)^{101} (1 - x + x^2)^{100} \] Rewriting \( (1 - x + x^2)^{100} \) as: \[ (1 + x)(1 + x^3)^{100} \] Expanding: \[ (1 + x) (1 + x^3)^{100} \] The coefficient of \( x^{50} \) in the expansion is determined by finding terms that yield \( x^{50} \) in the product. We require the coefficient of \( x^{50} \) in: \[ (1 + x) (1 + x^3)^{100} \] This is equivalent to: \[ \text{Coefficient of } x^{50} \text{ in } (1 + x^3)^{100} \] As \( 50 \) is not divisible by \( 3 \), no term contributes to \( x^{50} \), thus the coefficient is \( 0 \). Final Answer: \( \boxed{0} \).