  We investigate a class of diffusion-controlled reactions that are initiated
at the time instance when a prescribed number $K$ among $N$ particles
independently diffusing in a solvent are simultaneously bound to a target
region. In the irreversible target-binding setting, the particles that bind to
the target stay there forever, and the reaction time is the $K$-th fastest
first-passage time to the target, whose distribution is well-known. In turn,
reversible binding, which is common for most applications, renders theoretical
analysis much more challenging and drastically changes the distribution of
reaction times. We develop a renewal-based approach to derive an approximate
solution for the probability density of the reaction time. This approximation
turns out to be remarkably accurate for a broad range of parameters. We also
analyze the dependence of the mean reaction time or, equivalently, the inverse
reaction rate, on the main parameters such as $K$, $N$, and binding/unbinding
constants. Some biophysical applications and further perspectives are briefly
discussed.
