We deduce using the Ringel-Hall algebra approach explicit formulas for the
cardinalities of some Grassmannians over a finite field associated to the
Kronecker quiver. We realize in this way a quantification of the formulas
obtained by Caldero and Zelevinsky for the Euler characteristics of these
Grassmannians. We also present a recursive algorithm for computing the
cardinality of every Kronecker quiver Grassmannian over a finite field.