ACFT WAS CLRED FOR TKOF WHILE AN ACR WAS SHORT FINAL APCH RESULTING IN A GAR.

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DETROIT APCH CTL CLRED US FOR A VISUAL APCH TO RWY 21C AND SWITCHED US TO TWR CTL AT ABOUT 12 MI OUT. TWR CTLR CLRED US TO LAND ON RWY 21C AS SOON AS WE CHKED IN WITH HIM. WHEN WE WERE ABOUT 3 MI OUT; MY FO HEARD THE TWR CTLR CLR ANOTHER ACFT FOR TKOF FROM RWY 21C. I DID NOT HEAR THIS XMISSION. MY FO SAID HE SCANNED THE RWY LOOKING FOR AN ACFT THAT MIGHT HAVE BEEN HOLDING IN POS FOR TKOF BUT SAW NONE AND ASSUMED HE HAD MISUNDERSTOOD WHAT HE THOUGHT HE HEARD. HE DID SEE AN ACFT TAXIING NE ON THE 21C PARALLEL TAXIWAY BUT DISCOUNTED IT AS ONE CLRED TO TKOF BECAUSE OF HIS RELATIVELY FAR DISTANCE FROM THE RWY ENTRY POINT AND OUR RELATIVELY CLOSE-IN POS ON FINAL APCH. WHEN WE WERE AT ABOUT 300 FT AGL; (APPROX 1 MI ON FINAL) THAT ACFT TURNED ONTO THE RWY AND BEGAN HIS TKOF ROLL. WE INITIATED A GAR AND IN THE PROCESS DSNDED TO ABOUT 200 FT AGL. WE OVER FLEW THE OTHER ACFT CLOSE ENOUGH TO HEAR THE ENG NOISE FROM IT. THE ACFT WAS A MEDIUM SIZE TURBOJET (TWO ENGS) TYPE AND IDENT UNKNOWN. I DON'T THINK THAT I OR MY FO WAS AT FAULT IN THIS SITUATION. THAT IS; I BELIEVE WE DID EVERYTHING REQUIRED OF US BUT IF WE SIMPLY HAD QUERIED THE TWR CTLR ABOUT THE XMISSION THE FO THOUGHT HE HEARD; THE NEAR MISS COULD HAVE BEEN AVOIDED.