Sums, products, and double counting: name the objects first

If each object belongs to exactly one case, the total is the sum of the case sizes. Disjointness prevents duplication, and coverage prevents omissions. For a two-character code beginning with a letter or a digit, those first-character cases are disjoint if the alphabets are distinct. For people who know Python or Rust, the cases overlap when someone knows both, so adding the two totals counts that person twice. The word or alone does not justify addition.

A partition is therefore the natural structure behind the sum rule. It need not contain equally sized blocks. Counting by the first occurrence of a special event often creates disjoint cases even when the original description overlaps. Always state which unique feature assigns an object to a case. If the same object can be placed in two cases, either refine the classification or track the duplicated memberships explicitly.

The product rule counts outcomes produced by successive choices when each complete object has a unique construction history and each stage offers a fixed number of choices after every permitted earlier history. The actual options can depend on earlier choices: choosing two distinct people in order gives n choices followed by n minus one, even though the identity of the excluded second choice changes. Constant branch count, not probabilistic independence, is the relevant condition.

When branch counts vary, split into cases and sum the products along them. Suppose a password begins with A or B; after A there are three allowed suffixes, after B there are five. There are eight passwords, not two times an arbitrary suffix count. A tree is useful because each leaf is a complete outcome, but verify that distinct paths do not encode the same object if order is supposed to be forgotten at the end.

An incidence is a pair recording that one object participates in another: a student in a club, a vertex on an edge, or an element in a subset. Count the incidence set by fixing its first component and summing partners, then by fixing its second component and summing partners. Both sums equal the size of that same relation. This proves an identity without algebraic manipulation and often explains why a factor appears.

For a finite undirected graph without loops, count vertex-edge incidences. Summing over vertices gives the degree sum; summing over edges gives twice the edge count because each edge has two distinct endpoints. The even total then forces an even number of odd-degree vertices. If loops are allowed, define their degree contribution consistently as two. A counting argument is only as exact as its definition of an incidence and its treatment of exceptional objects.