  In this {\it case study}, we hope to show why Sheldon Axler was not just
wrong, but {\em wrong}, when he urged, in 1995: ``Down with Determinants!''. We
first recall how determinants are useful in enumerative combinatorics, and then
illustrate three versatile tools (Dodgson's condensation, the holonomic ansatz
and constant term evaluations) to operate in tandem to prove a certain
intriguing determinantal formula conjectured by the first author. We conclude
with a postscript describing yet another, much more efficient, method for
evaluating determinants: `ask determinant-guru, Christian Krattenthaler', but
advise people only to use it as a last resort, since if we would have used this
last method right away, we would not have had the fun of doing it all by
ourselves.
