1. **Problem:** A board of directors has 10 members. We want to elect a president, a vice president, and a secretary in that order. How many ways can this be done? 2. **Formula and rules:** This is a permutation problem because the order of selection matters (president, vice president, secretary are distinct positions). The number of permutations of selecting $k$ positions from $n$ members is given by: $$P(n,k) = \frac{n!}{(n-k)!}$$ 3. **Calculation:** Here, $n=10$ and $k=3$. $$P(10,3) = \frac{10!}{(10-3)!} = \frac{10!}{7!}$$ 4. **Simplify factorial:** $$\frac{10!}{7!} = 10 \times 9 \times 8 \times \cancel{7!} / \cancel{7!} = 10 \times 9 \times 8$$ 5. **Multiply:** $$10 \times 9 = 90$$ $$90 \times 8 = 720$$ 6. **Answer:** There are **720** ways to elect the president, vice president, and secretary in order.