A CL60 DEPARTED ON CLE RWY 24C WHEN CLRED FOR TKOF ON RWY 24R.

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WE RECEIVED OUR CLRNC TO ZZZ1. WE COMPLETED OUR CHKS AS WE TAXIED TO RWY 24R. MY FO WAS FOLLOWING GND'S TAXI INSTRUCTIONS AND MONITORING ON THE TAXI DIAGRAM. WE HAD RECEIVED A TAXI CLRNC TO RWY 24R AND WHILE ENRTE; HELD AT 2 INTXNS WHILE GND SEQUENCED OTHER TFC. AS WE APCHED RWY 24L; WE WERE GIVEN TKOF CLRNC TO RWY 24R WITH TFC ON A 5 MI FINAL. AS I TOOK THE RWY; I VISUALLY CLRED THE DEP PATH AND ASKED THE FO IF ALL WERE READY FOR TKOF. I THEN ADDED TKOF PWR. THE FIRST CALLS I HEARD WERE 80 KTS; V1; VR; AND ABOUT THAT TIME THE 'X' ON THE FAR END OF THE RWY CAME INTO SIGHT. THERE WERE NO MARKINGS AT TXWY S WHERE WE STARTED FROM. AS I LOOK BACK AND QUESTION MYSELF AND MY FO; MY RESPONSE WOULD BE FOR A BETTER TAXI BRIEFING -- ESPECIALLY IN THE EVENT OF CLOSED TXWYS AND RWYS; AS THEY WOULD PERTAIN TO US; COMMUNICATING AND UNDERSTANDING PRIOR TO TAXI IS A MUST. SUPPLEMENTAL INFO FROM ACN 643841: THIS WAS AN FAR PART 91 LEG FROM CLE TO ZZZ WITH NO PAX ON BOARD. THE COPLT IS TYPE RATED IN AND HAS BEEN FLYING THE CHALLENGER FOR SEVERAL YRS. RWY 24C WAS NOTAMED CLOSED AND ATIS ANNOUNCED THIS ALSO. WE OBSERVED A COMPANY AIRPLANE TAKE OFF ON RWY 24R WHILE WE WERE HOLDING SHORT OF RWY 24L. TWR'S CLRNC WAS TO TAXI ACROSS RWY 24L AND CLRED FOR TKOF RWY 24R WITH A HDG TO FLY. I CLRED FINAL OF TFC AND INITIATED THE LINE-UP CHKLIST. AS I FINISHED THE CHKLIST; THE COPLT WAS ADVANCING THE THROTTLES FOR TKOF. I WAS SOMEWHAT UNSURE OF WHAT RWY WE WERE ON AND BEGAN LOOKING FOR VISUAL CLUES THAT WERE ON THE CORRECT ONE. I DID NOT COMPREHEND THAT WE WERE ON THE WRONG RWY UNTIL LIFTOFF SPD; WHICH WAS TOO LATE TO ABORT. I STATED AT THAT TIME THAT WE JUST TOOK OFF ON THE WRONG RWY WHICH WAS RWY 24C. TWR CALLED US TO TELL US THE SAME. LOOKING BACK; AS SOON AS I BEGAN FEELING UNCERTAIN OF WHAT RWY WE WERE ON; I SHOULD HAVE CALLED FOR AN ABORT. I COULD HAVE PERFORMED THE LINE-UP CHKLIST SLOWER WITH PAUSES TO LOOK OUTSIDE AND MONITOR THE ACFT MOVEMENT. AN ATC CLRNC FOR US TO TAXI ACROSS RWY 24L AND RWY 24C; CLRED FOR TKOF RWY 24R MAY HAVE PREVENTED THIS INCIDENT. THERE IS NO DOUBT THAT THE COPLT MADE A MISTAKE AND THAT I DID NOT FIGURE THIS OUT IN TIME TO STOP IT. IF WHAT THE COPLT TELLS ME IS TRUE ABOUT THE MARKINGS ON THE CLOSED RWY NOT BEING VISIBLE; THIS IS A CONTRIBUTING FACTOR. ATC CLRNC WAS A CONTRIBUTING FACTOR SINCE IT IMPLIED THERE WAS ONLY A RWY 24L AND RWY 24R.