1. **Problem statement:** We have 20 books on a shelf, including 2 red-covered books that must not be placed next to each other. We want to find the number of ways to arrange all 20 books so that the two red books are not adjacent. 2. **Total arrangements without restriction:** The total number of ways to arrange 20 distinct books is $$20!$$. 3. **Arrangements with the two red books together:** Treat the two red books as a single combined item. Then we have $$19$$ items to arrange (the combined red pair + 18 other books). The number of ways to arrange these 19 items is $$19!$$. Since the two red books can be arranged among themselves in $$2!$$ ways, the total number of arrangements with the red books together is $$2! \times 19!$$. 4. **Arrangements with the two red books NOT together:** Use the complement rule: $$\text{Number of arrangements with red books not together} = 20! - 2! \times 19!$$ 5. **Simplify the expression:** Recall that $$20! = 20 \times 19!$$, so: $$20! - 2! \times 19! = 20 \times 19! - 2 \times 19! = (20 - 2) \times 19! = 18 \times 19!$$ 6. **Final answer:** The number of ways to arrange the 20 books so that the two red books are not together is: $$\boxed{18 \times 19!}$$