The very first thing to do when solving a counting problem isn't to reach for a formula — it's to figure out whether the situation calls for the multiplication rule or the addition rule. Get this judgment wrong, and no matter how carefully you calculate afterward, the answer comes out completely wrong. The two rules can look similar on the surface, but they apply to completely different situations.
The multiplication rule applies when two things happen one after another (in sequence). If you pick a drink and then pick a dessert, and there are m ways to pick the drink and n ways to pick the dessert, the total number of combinations is m×n. Why multiplication? Because every single time you pick a drink, an entire fresh set of n dessert choices opens up. It's as if that set of n gets multiplied m times over.
The addition rule applies when two cases can't happen at the same time (they're mutually exclusive). Think of a situation like "rolling a 3 on a die, or flipping heads on a coin" — you're choosing only one of the two. If case A can happen m ways and case B can happen n ways, and the two never overlap, the total is m+n.
The best way to avoid mixing up the two rules is to think about whether the situation connects with "and (then)" or with "or." "Pick a drink and pick a dessert" → the multiplication rule. "Take the bus or take the subway" → the addition rule. But the addition rule only adds the two cases directly when they don't overlap. If there is overlap, you need to subtract it once, just like the |A∪B|=|A|+|B|−|A∩B| formula you learned with sets.
A fun example of the two rules working together is counting shortest paths on a grid. When counting how many ways there are to reach an endpoint by moving only right and down, the very last step to reach any given point is either "from the left" or "from above" — one or the other (the addition rule). So the number of paths to each point is the sum of the path counts of the point to its left and the point above it, and if you fill in the whole grid this way, remarkably, the numbers you get are exactly Pascal's Triangle. In the end, counting paths and counting the coefficients in the expansion of (a+b)ⁿ share the exact same underlying counting method.
When studying this with kids, it helps to use a real situation and ask "is this an 'and' or an 'or'?" first. "Ways to pair 3 t-shirts with 2 pairs of pants" is an "and" (the multiplication rule, 3×2=6 ways), while "choosing either jjajangmyeon or jjamppong" is an "or" (the addition rule). On our activity page, you can watch the multiplication rule branch out like a tree with a menu-picking diagram, and watch the addition rule build Pascal's Triangle with shortest paths on a grid.