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Combinatorial analysis helps us determine the number of possible arrangements in a set when the order of the arrangements doesn't matter, often referred to as combinations. In our exercise, the customs official is not concerned with the order in which the three tablets are selected, only the content of the selection.

Consider a simple example: selecting a team of 3 people from a group of 10. It doesn't matter if Tom, Dick, or Harry are chosen first, second, or third; all that matters is whether they are on the team. The combination is represented using binomial coefficients, which look like \( \binom{n}{r} \) and read as 'n choose r'. These coefficients are a part of combinatorial analysis and are essential for solving our original problem.

The traveler's mix of vitamin and narcotic pills can be seen as two separate groups from which we are making selections. By determining the combinations possible with 6 narcotic and 9 vitamin pills when picking 3, we are applying combinatorial analysis to assess the probability of a specific outcome – in this case, getting caught.

The probability formula is a fundamental concept in statistics and probability theory, expressing the likelihood of a specific event occurring. It is represented as \( P(E) = \frac{Number\: of\: Desired\: Outcomes}{Total\: Number\: of\: Outcomes} \).

In our exercise, the event 'E' is the traveler getting arrested, contingent upon at least one narcotic pill being chosen. 'Desired outcomes' may feel misleading—certainly, the traveler doesn't desire arrest—but in probability terminology, these are simply the outcomes that fulfill the condition for event E to occur.

To determine the probability that an event will occur, calculate the possible ways that event can occur and divide it by the total number of possible outcomes. The challenge often lies in accurately determining these outcomes, as you must consider all possibilities that fit the criteria of the event.

Factorial notation is pervasive in combinatorial analysis and probability. The factorial of a non-negative integer 'n', denoted by \(n!\), is the product of all positive integers less than or equal to 'n'. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

Factorials are crucial when calculating permutations and combinations because they provide the total number of ways 'n' different items can be arranged, which is the foundation of combinatorial problems like our exercise. Importantly, \(0!\) is defined to be 1, which is consistent with the convention for an empty product. This convention ensures that formulae involving factorials remain valid even for situations where no items are selected, as seen in the \( \binom{9}{0} \) term of our desired outcomes calculation.