2023 AMC 10B Problem 12

When the roots of the polynomial \[P(x) = (x-1)^1(x-2)^2(x-3)^3 \cdots (x-10)^{10}\] are removed from the real number line, what remains is the union of 11 disjoint open intervals. On how many of those intervals is P(x) positive?

\textbf{(A)} 3 \qquad \textbf{(B)} 7 \qquad \textbf{(C)} 6 \qquad \textbf{(D)} 4 \qquad \textbf{(E)} 5

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.