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We present preliminary results of searches for charged B mesons decaying into the charmless three-body final states h{sup {+-}} h{sup {-+}} h{sup {+-}}, where h = {pi} or K, using 51.5 fb{sup -1} of data collected at the {Upsilon}(4S) resonance with the BABAR detector at the SLAC PEP-II asymmetric B Factory. No assumptions are made about intermediate resonances. We measure the branching fractions {Beta}(B{sup {+-}} {yields} K{sup {+-}} {pi}{sup {-+}} {pi}{sup {+-}}) = (59.2 {+-} 4.7 {+-} 4.9) x 10{sup -6} and {Beta}(B{sup {+-}} {yields} K{sup {+-}} K{sup {-+}} K{sup {+-}}) = (34.7 {+-} 2.0 {+-} 1.8) x 10{sup -6}, where the first error is statistical and the second error is systematic. In the same study, we do not observe significant signals for the final states B{sup {+-}} {yields} {pi}{sup {+-}} {pi}{sup {-+}} {pi}{sup {+-}} and B{sup {+-}} {yields} K{sup {+-}} K{sup {-+}} {pi}{sup {+-}}, and therefore provide the 90% confidence upper limits {Beta}(B{sup {+-}} {yields} {pi}{sup {+-}} {pi}{sup {-+}} {pi}{sup {+-}}) < 15 x 10{sup -6} and {Beta}(B{sup {+-}} {yields} K{sup {+-}} K{sup {-+}} {pi}{sup {+-}}) < 7 x 10{sup -6}.

Measurements of the Branching Fractions of Charmless Three-Body Charged B Decays.