  We address an idea of applying generalized entropies in counting problems.
First, we consider some entropic properties that are essential for such
purposes. Using the $\alpha$-entropies of Tsallis-Havrda-Charv\'{a}t type, we
derive several results connected with Shearer's lemma. In particular, we derive
upper bounds on the maximum possible cardinality of a family of $k$-subsets,
when no pairwise intersections of these subsets may coincide. Further, we
revisit the Minc conjecture. Our approach leads to a family of one-parameter
extensions of Br\'{e}gman's theorem. A utility of the obtained bounds is
explicitly exemplified.
