Question

Suppose that a saleswoman has to visit eight different cities. She must begin her

trip in a specified city, but she can visit the other seven cities in any order she

wishes. How many possible orders can the saleswoman use when visiting these

cities?

Answer 1

Since the first visiting city is fixed, for the second visit a saleswoman has 7 choises. Then for the third visit she has 6 choises, and so on. The eighth visit can be chosen uniquely. By Combinatorial Product Principle, the saleswoman can use $7\cdot 6\cdot\ldots \cdot1=7!=5,040$ possible orders.