Final answer:
The computer component that provides connections for all others is the motherboard. Microchips, or integrated circuits, greatly reduced the size and need for complicated wiring in computer systems, allowing for circuits in a compact space.
Step-by-step explanation:
The component within a computer system's case that provides connections for all other components is known as the motherboard. The microchip, specifically the type referred to as an integrated circuit (IC), plays a critical role in enabling these connections, as it incorporates a multitude of electronic circuits into a compact, silicon-based chip. This technological innovation is a key reason behind the shrinking of computer technology that once filled a whole room into something that can fit within a small box, such as a desktop computer, without the need for complicated wiring.
The other component mentioned, the vacuum tube, was used in earlier generations of computers before the advent of the integrated circuit. With the introduction of microchips, computers became more efficient, reliable, and capable of accommodating circuits in a significantly smaller space, which was a driving force in the proliferation of the personal computer.