CORP ACFT TKOF ON WRONG RWY AT GGG.

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I WAS THE CAPT AND INPUTTING OUR COURSE IN THE RNAV. IT WAS TSTMING AND RAINING IN ALL QUADRANTS AROUND THE ARPT. I WAS ALSO ANALYZING THE WX AND OUR BEST COURSE OF ACTION. MY COPLT; WHOSE SOLE PURPOSE IN THE AIRPLANE IS TO WORK THE RADIOS; RESPONDED TO ALL CALLS FROM THE TWR. GREGG CO TWR RESPONDED; 'YOU'RE CLRED TO TKOF ON 13.' I DIDN'T HEAR THE NUMBER OF THE RWY SAID IN THE XMISSION. IT WAS THEN TURNED TO MY COPLT AND ASKED; 'WE WERE CLRED TO TKOF ON 17; CORRECT?' HE VERIFIED BY SAYING; 'THIS IS CORRECT; 17.' (OUR DAY BEGAN AT EARLY AM; I COULD SEE HE WAS GROWING TIRED). I PROCEEDED TO TAXI ON RWY 17. NO OTHER AIRPLANES WERE AROUND AT THE TIME OF OUR TKOF. WE TOOK OFF ON RWY 17 AND THEN I HEARD TWR RESPOND; 'NO; YOU TOOK OFF ON 17; NOT 13; NO PROBLEM THOUGH; CONTACT DEP.' MY COPLT; ONCE AGAIN; DID NOT HEAR THIS XMISSION SO I RELAYED TO HIM THAT WE HAD TAKEN OFF ON THE WRONG RWY; THE ONE HE VERIFIED WAS CORRECT. I REALIZE IT IS MY RESPONSIBILITY; THE CAPT; TO TAKE THE RESPONSIBILITY OF MY NEGLIGENT COPLT AND MISTAKES. HIS 'TIRED' STATE OF AWARENESS SHOULD HAVE BEEN MORE APPARENT TO ME AND I SHOULD HAVE GOTTEN ON THE RADIO AND ASKED THEM MYSELF WHAT RWY TO TKOF (17 OR 13). I COUNTED ON AND ACCEPTED HIS RESPONSE TO MY QUESTION WITHOUT DOUBLECHKING. FROM NOW ON; I WILL TAKE IT UPON MYSELF; ALL QUESTIONS WHEN FLYING WITH HIM (OR ANY TIRED/UNATTENTIVE COPLT).