Summary: We introduce a call-by-name \(\lambda\)-calculus \(\lambda J_n\) with generalized applications which is equipped with distant reduction. This allows to unblock \(\beta\)-redexes without resorting to the standard permutative conversions of generalized applications used in the original \(\Lambda J\)-calculus with generalized applications of Joachimski and Matthes. We show strong normalization of simply-typed terms, and we then fully characterize strong normalization by means of a quantitative (i.e. non-idempotent intersection) typing system. This characterization uses a non-trivial inductive definition of strong normalization – related to others in the literature –, which is based on a weak-head normalizing strategy. We also show that our calculus \(\lambda J_n\) relates to explicit substitution calculi by means of a faithful translation, in the sense that it preserves strong normalization. Moreover, our calculus \(\lambda J_n\) and the original \(\Lambda J\)-calculus determine equivalent notions of strong normalization. As a consequence, \(\lambda J\) inherits a faithful translation into explicit substitutions, and its strong normalization can also be characterized by the quantitative typing system designed for \(\lambda J_n\), despite the fact that quantitative subject reduction fails for permutative conversions.