  The aim of this paper is to present a new proof of an explicit version of the
Berry-Ess\'{e}en type inequality of Bolthausen (Zeitschrift f\"ur
Wahrscheinlichkeitstheorie und Verwandte Gebiete, 66, 379--386, 1984). The
literature already provides several proofs of it using variants of Stein's
method. The characteristic function method has also been applied but led only
to weaker results. In this paper, we show how to overcome the difficulties of
this method by using a new identity for permanents of complex matrices in
combination with a recently proved inequality for the characteristic function
of the approximated distribution.
