FLT INSTRUCTOR AND STUDENT; IN SMA; TAKE OFF FROM WRONG RWY.

Narrative

I HAD RECEIVED A TWR ENRTE CLRNC AT SANTA BARBARA ARPT FOR RWY 25 TO TORRANCE ARPT (TOA). I WAS THEN CLRED TO TAXI TO RWY 15L TO DO MY RUN-UP. I WAS TRYING TO HELP MY STUDENT SET UP FOR THE IFR CLRNC AND IN THE PROCESS WAS LETTING HIM DO THE RADIOS. WHEN WE WERE READY TO DEPART; HE CONTACTED SBA TWR AND RECEIVED A CLRNC. I THOUGHT I HEARD 'CLRED TO TKOF RWY 15L;' WHICH WE WERE ON AND MY STUDENT BEGAN TO TKOF ON THAT RWY. IT SEEMED ODD AT THE TIME; BUT I THOUGHT THEY WERE TRYING TO EXPEDITE OUR DEP ON RWY 15L SO WE WOULDN'T HAVE TO CIRCLE AROUND S TO GO TO TORRANCE. AFTER LIFT-OFF WE WERE INFORMED THAT WE USED THE WRONG RWY. LOOKING BACK; IT MAKES NO SENSE FOR THEM TO HAVE ME TAKE OFF FROM RWY 15L WHEN OUR IFR CLRNC WAS FOR RWY 25. I STILL BELIEVE I HEARD RWY 15L; BUT I NOW REALIZE THAT I WAS DOING TOO MANY TASKS AT ONE TIME. WE WERE ALSO IN A HURRY TO GET THE PLANE BACK. SO; MOST LIKELY I HEARD WHAT I WANTED TO HEAR INSTEAD OF WHAT THE ACTUAL CLRNC WAS. TO AVOID THIS FROM HAPPENING AGAIN; I WILL MAKE SURE THAT I DO ONE TASK AT A TIME AND WOULD RECOMMEND THAT TO OTHER INSTRUCTORS.