BE76 TRAINING ON INST APCHS. ELECTED TO MAKE A TOUCH AND GO AFTER ILS RWY 35R. ON ROLLOUT TRAINEE RAISED GEAR LEVER VERSUS FLAPS. MAIN GEAR REMAINED DOWN BUT NOSE GEAR RETRACTED RESULTING IN DAMAGE TO NOSE AND BOTH PROPS. NO INJURIES.

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I WAS ACTING FLT INSTRUCTOR; INSTRUCTING INST APCHS. MY STUDENT HAS HER MULTI-PVT AND 60 HRS IN THE TYPE: BE76. WE WERE FLYING AN ILS RWY 35R INTO WILL ROGERS WORLD ARPT. MY STUDENT FLEW THE APCH VERY WELL AND THERE WAS NO INDICATIONS OF ANY PROB. AT DECISION HT MY STUDENT TOOK HER 'HOOD' OFF TO MAKE A TOUCH AND GO; AS SHE WAS TOLD TO DO. AFTER TOUCHDOWN WE ROLLED FOR A SHORT WHILE AND MY STUDENT REACHED FOR THE FLAPS BUT RAISED THE GEAR INSTEAD. AS SHE REACHED FOR THE GEAR HANDLE I TRIED TO STOP HER BUT SHE WAS QUICKER. SHE REALIZED HER MISTAKE AND PUT THE GEAR HANDLE BACK INTO THE DOWN POS. HOWEVER; THE OVER CENTER LOCK WAS ALREADY DISENGAGED AND THE NOSE SLOWLY SUNK TO THE GND. THE NOSE; BOTH PROPS WERE DAMAGED. NEITHER ME NOR MY STUDENT WAS INJURED AND WE WALKED AWAY FROM THE ACFT. I DO NOT SEE ANYTHING THAT I COULD HAVE DONE AS A FLT INSTRUCTOR OTHER THAN BEING QUICKER. SUPPLEMENTAL INFO FROM ACN 361496: WHILE MAIN GEAR IS COMPRESSED NOSE GEAR SHOULD NOT BE ALLOWED TO RETRACT. SQWAT SWITCH SHOULD BE RE-ENGINEERED. DUE TO STUDENT ERROR -- INSTRUCTOR WAS UNABLE TO REVERSE PROB. THEREFORE HE SHOULD NOT HAVE TO DO A 609 RIDE BUT RATHER THE STUDENT.