the addition rule for counting

Imagine you want a snack and the kitchen has 4 kinds of fruit and 3 kinds of biscuit, and you will pick exactly one snack. How many choices do you have? You are not doing two steps in a row; you are choosing between two separate piles. The answer is simply 4 + 3 = 7. This is the addition rule: when you split a count into cases that never overlap, you add the cases up.

Formally, if a task can be done in one of several mutually exclusive ways — say case A can happen in m ways and case B in n ways, and no single outcome belongs to both A and B — then the total number of ways is m + n. 'Mutually exclusive' is the crucial word: the cases must not share any outcome, or you would count something twice. The clean way to use the rule is to chop your problem into non-overlapping buckets, count each bucket, and sum. For example, the number of 2-digit numbers that are even (45 of them) plus the number that are odd (45) gives all 90 two-digit numbers, because no number is both even and odd.

If the cases CAN overlap, plain addition over-counts the shared part, and you must subtract it back out: |A or B| = |A| + |B| - |A and B|. That correction is the start of the inclusion-exclusion principle. So the addition rule in its simple form is really the special case of inclusion-exclusion where the overlap is empty. Together with the multiplication rule, addition lets you decompose almost any counting problem: multiply within a sequence of steps, add across separate cases.