A chemical solution contains $N$ molecules of type $\mathrm{A}$ and $M$ molecules of type $\mathrm{B}$. An irreversible reaction occurs between type $\mathrm{A}$ and type $\mathrm{B}$ molecules in which they bond to form a new compound $\mathrm{AB}$. Suppose that in any small time interval of length $h$, any particular unbounded $\mathrm{A}$ molecule will react to any particular unbounded $\mathrm{B}$ molecule with probability $\theta h + o(h)$ where $\theta$ is a reaction rate. Let $X(t)$ denote the number of unbounded $\mathrm{A}$ molecules at time $t$. Model $X(t)$ as a pure death process by specifying parameters.
The answer is
$k\big(M - (N - k)\big)\theta$
for $k = 0, 1, 2, \dots, N$.
I am unsure about what the expression $k\big(M - (N - k)\big)$ represents and would appreciate if someone could explain the rationale behind it to me. The way I approached this question was I took k to represent the number of $A$ molecules remaining. If we want $P (X(t) = k)$, it is equivalent to saying $N - k$ molecules died. If you subtract that from $M$, that is the remaining number of $\mathrm{B}$ molecules remaining to react with. Thus, multiplying that by theta should give you the rate. However, the solution does not match my rationale and adds in a $k$.