1. Apparatus for counting and separating nested articles as such articles move along a predetermined path, comprising a shaft mounted to rotate and move between first and second positions; a counting wheel secured to the shaft and having teeth about the wheel''s rim some teeth being long relative to other teeth, and with the shaft in the first position said individual teeth engaging individual article lips to count the moving articles, whereby the wheel and shaft rotate; means for monitoring the rotation of the wheel and shaft and, when an adjacent short tooth and long tooth each engage adjacent article lips, for moving the shaft to the second position to disengage the short tooth from one article lip but maintain engagement between the long tooth and the other article lip, whereby said predetermined number of articles have passed the wheel and been counted; and means for separating the counted articles from the remainder of the nested articles while the shaft is in the second position.

5. Apparatus for counting and separating nested articles or the like as such articles move along a predetermined path, comprising a movable carrier assembly having a homing position from which it moves and returns, said assembly including a hollow shaft mounted on the assembly so that said shaft rotates and can be moved along its longitudinal axis between first and second positions; a counting wheel secured to the shaft and having teeth about the wheel''s rim, some teeth being long relative to other teeth, and when the shaft is in the first position individual teeth engaging individual article lips to count the moving articles, whereby the wheel and shaft rotate; port means in at least one long tooth that is adjacent a short tooth, said port means being in communication with the hollow shaft; a supply of compressed gas coupled to one end of the hollow shaft; means coupled to the other end of the shaft and adapted to move the shaft between the first and second positions; means for monitoring the rotation of the wheel and shaft and, when a predetermined number of articles have been counted by the wheel and when the long tooth with said port means and its adjacent short tooth each engage adjacent article lips, for actuating the shaft moving means to move the shaft from the first position to the second position, whereby said predetermined number of articles have passed the wheel and been counted; means for holding the wheel and shaft stationary while in the second position, whereby the movable carrier assembly leaves home position and moves with the moving nested articles; means for sensing when the shaft and wheel are in the second position, for releasing from the gas supply a jet stream which flows through the shaft and port means and separates counted articles from the remainder of the nested articles, and for actuating the shaft moving means to move the shaft from the second to the first position; and means for returning the carrier assembly to home position when the shaft is returned to the first position.