1. **Problem:** Calculate the value of $20P_3$. 2. **Formula:** The permutation formula is given by $$nP_r = \frac{n!}{(n-r)!}$$ where $n$ is the total number of items, and $r$ is the number of items to arrange. 3. **Apply the formula:** $$20P_3 = \frac{20!}{(20-3)!} = \frac{20!}{17!}$$ 4. **Simplify factorials:** $$\frac{20!}{17!} = 20 \times 19 \times 18$$ 5. **Calculate the product:** $$20 \times 19 = 380$$ $$380 \times 18 = 6840$$ 6. **Answer:** $$20P_3 = 6840$$ This means there are 6840 ways to arrange 3 items out of 20 in order.