Final answer:
The hardware that triggers some operation after a certain programmed count is the programmable interval timer (PIT).
Step-by-step explanation:
The hardware that triggers some operation after a certain programmed count is the programmable interval timer (PIT). The PIT is a device found in computer systems that can be programmed to generate interrupts at specific time intervals or after a certain number of counts. It is commonly used for implementing system timekeeping, generating periodic interrupts for multitasking operating systems, and controlling delays in various applications.