1. **State the problem:** We need to find the volume of a cylindrical pipe excluding the core. The pipe consists of an outer cylinder with radius $12$ cm and height $20$ cm, and an inner core cylinder with radius $4$ cm and height $20$ cm. 2. **Formula for volume of a cylinder:** $$V = \pi r^2 h$$ where $r$ is the radius and $h$ is the height. 3. **Calculate the volume of the outer cylinder:** $$V_{outer} = \pi \times 12^2 \times 20 = \pi \times 144 \times 20 = 2880\pi$$ 4. **Calculate the volume of the inner core cylinder:** $$V_{inner} = \pi \times 4^2 \times 20 = \pi \times 16 \times 20 = 320\pi$$ 5. **Calculate the volume of the pipe excluding the core:** $$V = V_{outer} - V_{inner} = 2880\pi - 320\pi = (2880 - 320)\pi = 2560\pi$$ 6. **Final answer:** $$V = 2560\pi \text{ cm}^3$$ This is the exact volume of the pipe excluding the core, without rounding.