the fundamental counting principle (multiplication rule)

Suppose you are getting dressed and you own 3 shirts and 4 pairs of trousers. How many different outfits can you make? You do not have to list them all to know the answer: for each of the 3 shirts you can pair any of the 4 trousers, so there are 3 times 4 = 12 outfits. That little leap, from 'choices made in steps' to 'multiply the number of choices', is the most important idea in all of counting.

The principle says: if a process consists of a sequence of steps, where the first step can be done in n1 ways, the second in n2 ways, the third in n3 ways, and so on, AND the number of ways at each step does not depend on which choices you made earlier, then the whole process can be done in n1 times n2 times n3 times ... ways. The key picture is a branching tree: the first choice splits into n1 branches, each of those splits into n2 branches, and the leaves at the bottom are the total outcomes, which is just the product of the branch counts. For example, a 4-digit PIN where each digit is 0 to 9 has 10 times 10 times 10 times 10 = 10^4 = 10000 possibilities.

The hidden condition 'the count at each step does not depend on earlier choices' is what people get wrong. The COUNT must be steady even if the specific options change. If you pick a 3-letter code with no repeated letters, the first letter has 26 options, the second has 25 (whichever you used is gone), the third has 24 — the counts 26, 25, 24 are fixed regardless of which letters you happened to pick, so you may multiply to get 26 times 25 times 24. This is exactly the engine behind permutations and combinations.