The coefficient of $x^4$ in the expansion of $(1 + x + x^2)^7$ is
Step-by-Step Solution
Key Concept: When expanding $(1+x(1+x))^y$, terms contributing to $x^4$ come from multiple binomial terms with different powers of $x$.
The coefficient of $x^4$ comes from expanding $(1+x(1+x))^y$ using the binomial theorem. When we expand $^yC_2x^2(1+x)^2 + ^yC_3x^3(1+x)^3 + ^yC_4x^4(1+x)^4$, the $x^4$ term arises from: $^yC_2 \cdot x^2 \cdot 1$, $^yC_3 \cdot x^3 \cdot x$, and $^yC_4 \cdot x^4 \cdot 1$. Thus, the coefficient of $x^4$ is $^yC_2 + ^yC_3 + ^yC_4$.
Correct Answer: 3