A surjection is a function that misses none of the target values, so missing-value events can be excluded. A small computation will anchor the general statement before the abstraction takes over.
Set-up
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 definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).
The calculation
The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.
What survives abstraction
The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.
The boundary
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.
I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.