Proposition: Uniform initial placement and active hits

Suppose there are $U=N_W-C$ unscarred coordinates, of which $A$ are active. Choose $k\le U$ new scars uniformly without replacement from those $U$ coordinates. The number $J$ of active coordinates hit has law

on its feasible integer range, and

The scar count increases by $k$, while the active count decreases by $J$.

Proof. Count the $k$-subsets with exactly $j$ active elements. For the expectation, each active coordinate is selected with probability $k/U$. Sum its indicator.