(Refer to the discussion after Example 4 . ) A salesperson...

(Refer to the discussion after Example $4 .$ ) A salesperson must travel to 3 of 7 cities. Direct travel is possible between every pair of cities. How many arrangements are there in which the salesperson could visit these 3 cities? Assume that traveling a route in reverse order constitutes a different arrangement.

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Key Concepts

Multiplicative Counting Principle

This foundational principle in counting states that if one event can occur in a certain number of ways and another independent event can occur in another number of ways, the total number of outcomes is found by multiplying the number of ways each event can occur. This principle is often applied when a problem requires sequential choices, such as first selecting items (using combinations) and then arranging them (using permutations).

Permutations

Permutations refer to the arrangements of items where the order does matter. Once a subset has been chosen using combinations, permutations are used to count the number of possible orders to arrange these items. In problems involving routes or sequences, recognizing when to use permutations is crucial to count distinct ordered arrangements correctly.

Combinations

This concept involves selecting a subset of items from a larger set, where the order of selection does not matter. In counting problems, combinations are used to determine the number of ways to choose items without considering the sequence in which they are picked, typically computed using binomial coefficients (often written as 'n choose r').

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