In charmless $B\to PPP$ decays, where $P$ is a pseudoscalar meson, there are six possibilities for the symmetry of the final state. In this paper, for $P=\pi ,K$, we examine the properties of the fully-symmetric final state. We present expressions for all 32 $B\to PPP$ decay amplitudes as a function of both SU(3) reduced matrix elements and diagrams, demonstrating the equivalence of diagrams and SU(3). We also give 25 relations among the amplitudes in the SU(3) limit, as well as those that appear when the diagrams $E/A/PA$ are neglected. In the SU(3) limit, one has the equalities $\sqrt{2}\mathcal{A}(B^{+}\to K^{+}{\pi }^{+}{\pi }^{-}{)}_{\text{FS}}=\mathcal{A}(B^{+}\to K^{+}{K}^{+}{K}^{-}{)}_{\text{FS}}$ and $\sqrt{2}\mathcal{A}(B^{+}\to {\pi }^{+}{K}^{+}{K}^{-}{)}_{\text{FS}}=\mathcal{A}(B^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}{)}_{\text{FS}}$, where FS denotes the fully-symmetric final state. These provide good tests of the standard model that can be carried out now by the LHCb Collaboration.

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