SA340 CREW TAXIING FOR DEP AT CAK UNDERSTOOD AND READ BACK TAXI TO RWY 01 INSTEAD OF RWY 19.

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FO CALLED FOR TAXI WITH INFO. GND REPLIED TAXI TO RWY 1. FO READ BACK TAXI TO RWY 1 AND I PROCEEDED TO TAXI ON 'A' TXWY ACROSS RWY 14/32 TO HOLD SHORT OF RWY 1. RWY 14 WAS IN USE FOR DEP AND NO CONFLICT WHATSOEVER OCCURRED. I QUESTIONED THE FO ON US USING RWY 1 AND HE CALLED FOR THE WINDS. GND CTL'S REPLY WAS 200 DEGS AT 13 KTS. WE SWITCH TO TWR AND I (CAPT) SAID WE CAN'T USE RWY 1; BECAUSE OF THE TAILWIND. HE SAID THAT WE WERE SUPPOSED TO TAXI TO RWY 19. I REPLIED THAT WE HEARD RWY 1. WE THEN RECEIVED NEW INSTRUCTIONS TO TAXI TO RWY 19 AND DEPARTED WITHOUT INCIDENT. CONTRIBUTING FACTORS WERE FATIGUE. WE HAD BEEN ON DUTY SINCE XA:00 LOCAL TIME AND HAD NOT HAD TIME TO EAT ANY FOOD ALL DAY DUE TO MAINT DELAYS AT A PREVIOUS OUTSTATION. ALSO RELATED; THE TWR CTLR APPEARED TO HAVE BEEN WORKING ALL FREQS AT THE TIME AND WAS DISTRACTED BY CLRNC CALLS/READBACKS AND INBOUND TFC. ALSO; WE WERE RUNNING BEHIND BEING PUSHED BY OUR RAMP PERSONNEL TO TURN THE ACFT QUICKLY. I ALSO THOUGHT THAT PERHAPS THE WIND HAD SHIFTED AND THUS THE RWY IN USE SWITCHED. WE DID GET THE WINDS FROM GND AND STARTED OUR TAXI TO RWY 1. WE PROBABLY COULD HAVE PREVENTED THE SITUATION BY QUESTIONING THE USE OF RWY 1 (OUR PERCEPTION) EARLIER. THERE IS ALSO A POSSIBILITY THE CTLR LET OFF ON THE MIC SWITCH EARLY OR WAS BLOCKED ON THE NINER PART OF RWY 19 (READ AS 'ONE NINER'). THIS RWY COULD BE EASILY CONFUSED IF THE 'NINER' PART IS BLOCKED. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR STATED THAT ON REACHING RWY 01; THEY IMMEDIATELY REALIZED THEY WERE AT THE INCORRECT RWY. THEY WERE INFORMED BY THE CTLR THAT A CLRNC TO RWY 19 HAD BEEN ISSUED.