The number of different terms in the expansion of $(1 - z^{30})\left(1 + z + x^2\right)^{30}$ is
Step-by-Step Solution
Key Concept: When expanding a product, the total number of distinct terms equals the sum of terms from each factor when the powers don't overlap.
The expansion $(1-x)(1-x^2)^{200}$ can be written as $(1-x) \cdot [(1-x^2)^{200}]$. The expansion of $(1-x^2)^{200}$ has terms with even powers from $0$ to $400$, giving $201$ terms. When we multiply by $(1-x)$, we get two sets of terms: the original terms from $(1-x^2)^{200}$ contributing with powers $0, 2, 4, \ldots, 400$, and the terms multiplied by $(-x)$ contributing with powers $1, 3, 5, \ldots, 401$. No overlap occurs between even and odd powers. Therefore, the total number of required terms is $201 + 201 = 402$.
Correct Answer: 402