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When faced with the task of selecting items from a larger group without caring for the order in which they're chosen, the combination formula comes into play. It's a fundamental concept in combinatorics, a branch of mathematics dedicated to counting, arranging, and grouping items.

The formula is expressed as: \( C(n, k) = \frac{n!}{k! (n - k)!} \), where \( n \) is the total number of items to choose from, \( k \) is the number of items to be chosen, and the symbol \( ! \) denotes a factorial. To provide a practical understanding:

• Imagine you're at a buffet with 8 dishes, and you want to try only 2. How many ways can you create your plate?
• Or, picture a soccer team of 11 players, but only 5 can be forwards. The combination formula will help you determine the number of possible forward lineups.

Such problems are efficiently solved using the combination formula because it effortlessly calculates the possible groupings, omitting permutations or different orderings which do not matter in these scenarios.

Factorial notation is a compact way to represent the multiplication of a series of descending natural numbers and is essential in calculations involving permutations and combinations. The notation \( n! \) (read as 'n factorial') means multiplying all whole numbers from \( n \) down to 1. For example:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)

Notably, by convention, \( 0! \) is defined as 1, which often confuses beginners. Factorials grow rapidly with larger numbers, making manual calculations unwieldy, which is why understanding factorial notation is crucial for simplifying such calculations, especially in combinatorial problems.

In probability, events are considered independent when the outcome of one event does not influence the outcome of another. This concept is pivotal for analyzing scenarios where multiple choices or occurrences are involved.

For example, rolling a die and flipping a coin are independent events; the result of the coin toss does not affect what number the die will show. Mathematically, if the events are independent, the probability of both occurring is the product of their individual probabilities: \( P(A \text{ and } B) = P(A) \times P(B) \).
This principle also extends to combination problems. In our exercise, selecting primary and secondary routes are independent choices—the selection of one does not restrict the other. Hence, we multiply the number of combinations of each to find the total possibilities.

Solving mathematical problems effectively often requires a structured approach. The step-by-step method showcased in the exercise is a classic example of mathematical problem-solving.
It starts by defining the problem clearly, then breaking down the problem into manageable chunks — in this case, separate calculations for primary and secondary route combinations. Afterward, it proceeds to apply appropriate mathematical formulas and principles, such as the combination formula and the concept of independent events. Lastly, the step-by-step process concludes with carrying out the calculations to arrive at the final answer.
Mathematical problem-solving isn't just about getting the answer, but also understanding and applying concepts to new and varied problems, reinforcing the learning and making it applicable to real-world situations.