1. **Problem:** Compute $P_6^3$ (the number of permutations of 6 items taken 3 at a time). 2. **Formula:** The permutation formula is $$P_n^r = \frac{n!}{(n-r)!}$$ where $n$ is the total number of items, and $r$ is the number of items chosen. 3. **Apply the formula:** $$P_6^3 = \frac{6!}{(6-3)!} = \frac{6!}{3!}$$ 4. **Calculate factorials:** $$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$$ $$3! = 3 \times 2 \times 1 = 6$$ 5. **Simplify:** $$P_6^3 = \frac{720}{6} = 120$$ 6. **Answer:** There are $120$ permutations of 6 items taken 3 at a time.