B727. TWR CTLR CLAIMED CREW TOOK OFF WITHOUT TKOF CLRNC.

Narrative

WE WERE TAXIING TO RWY 25R AT LAX ARPT. WE WERE APCHING THE LAST INTXN BEFORE THE END OF THE RWY. I TURNED THE RADIO TO TWR FREQ AND THE TWR CTLR IMMEDIATELY; WE BELIEVE; CLRED US FOR TKOF AT THE INTXN. WE ASKED THE FE IF WE HAD THE PERFORMANCE; AND HE SAID YES; BUT THE CAPT SAID REQUEST FULL LENGTH. I REQUESTED FULL LENGTH; AND AT THIS POINT WE BELIEVE THE TWR CLRED US FOR TKOF FULL LENGTH. THE CAPT CALLED FOR FINAL ITEMS (PRE TKOF CHK) AND SAID 'YOUR THROTTLES.' WE LINED UP ON THE RWY; APPLIED PWR AND TOOK OFF. UPON CONTACTING DEP CTL WE WERE GIVEN A HDG; AND TOLD WE WERE TO CONTACT A PHONE NUMBER ABOUT A POSSIBLE PLTDEV. THE TWR SAID WE HAD BEEN CLRED INTO POS RWY 25R; NOT CLRED FOR TKOF. THERE WAS NO RWY OR TFC CONFLICT. CONTRIBUTING FACTORS: 1) WE WERE ON THE LAST DAY OF A 3 DAY TRIP. ALL 3 MORNINGS WE GOT UP AROUND XA00 AND THE PREVIOUS DAY WE HAD A DIVERSION TO PORTLAND ON OUR WAY TO SEATTLE. ON THAT DAY WE SPENT CLOSE TO 10 HRS IN THE COCKPIT WITH OVER 7 HRS OF FLT TIME. THEY PUSHED OUR DEP BACK 1/2 HR SO WE WOULD BE LEGAL ON THE DAY WE HAD THE INCIDENT (WE WOULD HAVE BEEN 9 1/2 HRS IN 24) AND I WAS NOT JUST TIRED; BUT TRULY FATIGUED. THE REST OF THE CREW ALSO WAS VERY TIRED. THE FIRST NIGHT OF THIS TRIP WAS ALSO A SHORT NIGHT (ABOUT 9 HRS AT THE HOTEL WITH AN EARLY SHOW). 2) TWR CLRNC; WE HAD JUST FINISHED OUR CHKLIST AS WE SWITCHED TO TWR; AND THE IMMEDIATE TKOF CLRNC AT THE INTXN CAUGHT US OFF GUARD (DISRUPTED OUR FLOW) AND PUT US IN THE MIND SET OF FULL LENGTH TKOF AT THE END OF THE RWY.