A LOW TIME CFI BECAME CONFUSED WHEN HIS STUDENT LINED UP ON THE RWY BUT DID NOT TAKE OFF. THE LOW TIME CFI DIRECTED THE STUDENT TO TAKE OFF. THOUGH THE STUDENT SAID THEY HAD NOT BEEN CLRED; HE THEN COMPLIED WITH HIS CFI. AS THEY ROTATED; TWR CALLED AND SAID THEY HAD NOT BEEN CLRED FOR TKOF.

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I AM A CFI AND WAS FLYING WITH A STUDENT. WE TAXIED FROM TIEDOWN AT HPN; DID OUR RUNUP; AND WE WERE #1 IN SEQUENCE FOR DEP ON RWY 16. THE STUDENT WAS HANDLING ALL COMS WITH ATC. THE STUDENT TALKED TO THE TWR AND WE TAXIED ONTO THE RWY. AFTER POSITIONING THE ACFT FOR TKOF; THE STUDENT SAT THERE WITHOUT INITIATING THE TKOF. I PROMPTED THE STUDENT TO BEGIN THE TKOF. THE STUDENT INDICATED THAT HE DIDN'T REMEMBER HEARING THAT WE WERE 'CLRED FOR TKOF.' I INDICATED THAT WE WERE IN FACT CLRED AND THAT HE SHOULD INITIATE THE TKOF; WHICH HE DID. JUST AS THE AIRPLANE LIFTED OFF OF THE RWY; ATC CALLED US WITH A 'CLRED FOR TKOF.' I RESPONDED TO ATC BY REPEATING THAT WE WERE 'CLRED FOR TKOF.' APPROX 30 SECONDS AFTER THIS EXCHANGE WITH ATC; WHEN WE WERE APPROX 200-300 FT ABOVE THE RWY; THE CTLR CALLED US AND SAID THAT SHE DIDN'T REMEMBER CLRING US FOR TKOF PRIOR TO OUR INITIATING OUR TKOF ROLL. SHE FURTHER INDICATED THAT A KING AIR HAD BEEN CLRED ACROSS THE FAR END OF THE RWY. I DID SEE THE KING AIR AS WE LIFTED OFF FROM THE RWY BUT IT WAS ALREADY OFF OF RWY 16 AND ONTO A TXWY ON THE E SIDE OF THE ARPT. I RESPONDED TO HER INQUIRY BY INDICATING THAT I WAS PROBABLY AT FAULT. SHE DIDN'T SAY ANYTHING FURTHER.