Counting incidences first by vertices and then by edges gives the handshaking lemma. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.
Statement
Enumerative combinatorics turns a finite set \(\Omega\) into several reversible descriptions. Binomial coefficients \(\binom nk\) appear whenever a choice forgets order but remembers size.
The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.
Worked algebra
Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.
Conceptual compression
The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.
Caveat
A formula with the correct magnitude can still count the wrong objects. The proof must explain whether order, repetition, labels, and empty parts are allowed.
The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.