  Under flavor $SU(3)$ symmetry (SU(3)$_F$), the final-state particles in $B\to
PPP$ decays ($P$ is a pseudoscalar meson) are treated as identical, and the
$PPP$ must be in a fully-symmetric (FS) state, a fully-antisymmetric (FA)
state, or in one of four mixed states. In this paper, we present the formalism
for the FA states. We write the amplitudes for the 22 $B\to PPP$ decays that
can be in an FA state in terms of both SU(3)$_F$ reduced matrix elements and
diagrams. This shows the equivalence of diagrams and SU(3)$_F$. We also give 15
relations among the amplitudes in the SU(3)$_F$ limit, as well as the
additional four that appear when the diagrams $E$/$A$/$PA$ are neglected. We
present sets of $B \to PPP$ decays that can be used to extract $\gamma$ using
the FA amplitudes. The value(s) of $\gamma$ found in this way can be compared
with the value(s) found using the FS states.
