The addition principle is a basic counting principle.
Intuitively, it tells us that if a situation can occur in different ways, and another situation (exclusive from the first one) can occur in different ways, then there are ways in which the first situation or the second one can occur.
Notice that “or” is the keyword for addition. It tells us that we don’t care specifically about which situation occurs; we only care if either of them occurs.
Compare this with multiplication, which uses “and”.
Mathematically, if and are two finite disjoint sets, then:
Let’s say we’re shopping for shirts at two stores.
In one store, we can choose from different colours of shirts. In the other store, we can choose from different colours.
In total, we can choose from different colours.