AMC 12 2023b Problem 6

When the roots of the polynomial

$$P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}$$

are removed from the number line, what remains is the union of 11 disjoint open intervals. On how many of these intervals is $P(x)$ positive?

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