### Abstract

This entry shows that the falling factorial of a sum can be computed
with an expression using binomial coefficients and the falling
factorial of its summands. The entry provides three different proofs:
a combinatorial proof, an induction proof and an algebraic proof using
the Vandermonde identity. The three formalizations try to follow
their informal presentations from a Mathematics Stack Exchange page as
close as possible. The induction and algebraic formalization end up to
be very close to their informal presentation, whereas the
combinatorial proof first requires the introduction of list
interleavings, and significant more detail than its informal
presentation.