Simple Counting Techniques and the Multiplication Principle

Think about outfits. Five shirts and three pairs of trousers don't give five plus three outfits. Each shirt can be worn with all three trousers, so you get five separate blocks of three outfits each: 5 x 3 = 15. Adding a shirt to the closet doesn't add one outfit, it adds a whole new block of three, because that shirt still has to be paired with every pair of trousers you own. That's the whole idea behind the multiplication principle: when a task breaks into stages, and the choice at each stage doesn't depend on what happened before, multiply the number of options at each stage together to get the total count.

The mechanism is just repeated pairing. Picture filling in slots left to right. For a 4 character code made of one letter followed by three digits, with repeats allowed, the letter slot has 26 options and each digit slot has 10 options, independent of the others. The total is 26 x 10 x 10 x 10 = 26,000. Each slot's count gets multiplied in, because every choice in one slot can combine with every combination of choices in the rest.

Forbidding repeats changes the arithmetic but not the logic. Once a letter or digit is used, it leaves the pool, so each slot after the first has one fewer option than it would with repeats allowed. This is the bridge into permutations: arranging n people into distinct positions with no repeats is exactly this shrinking-pool multiplication, stage by stage, until every position is filled.