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In combinatorics, we often need to determine the number of ways to pick a subset from a larger set when the order of selection does not matter. This is where the combination formula comes in handy. The general formula is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n\) represents the total number of items, and \(k\) represents the number of items to choose. For instance, in the given exercise, the FedEx driver needs to choose 6 locations out of 9, so we use \(\binom{9}{6}\). Combinations are different from permutations because the order in which the items are selected doesn’t matter. Imagine having 3 colored balls (red, blue, green); selecting red and then blue is the same as selecting blue and then red in combinations.

Factorials play a crucial role in calculating combinations. A factorial of a non-negative integer \(n\), denoted as \(n!\), is the product of all positive integers less than or equal to \(n\). For instance, \(4! = 4 \times 3 \times 2 \times 1 = 24\). Factorials grow very quickly with larger values of \(n\), which is why they are extremely useful in combinatorics. In the provided exercise, we use factorials to compute the number of ways to schedule deliveries. The factorial expressions simplify the combination formula calculation. For example, \(9! = 362,880\), \(6! = 720\), and \(3! = 6\). These values are then plugged into the combination formula to get the final result.

Although the given exercise focuses on combinatorics, understanding combinations helps in calculating probabilities as well. Probability measures how likely an event is to occur, and is often expressed as a fraction between 0 and 1. When calculating probabilities, we often use combinations to determine the number of favorable outcomes out of the total possible outcomes. For instance, if we want to determine the probability that a person picks 6 specific locations out of 9 without concern for order, we can use the combination formula to find the number of possible sets of locations. By comparing this to the total number of possible outcomes, we can calculate the probability. Thus, a solid understanding of combinations directly supports understanding and solving probability problems.