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If you would have to pick 4 armies to function as a sample or cross section of all that W40K has to offer, which would you pick?

These four would have to represent the range of the universe fluff wise, so Imperium, Xenos, etc. But they would also have to show different play styles/army build styles, so shooty, assaulty, horde, elite. Vanilla Space Marines would be the obvious first choice and probably Ork or Tyranids as second.

What other armies to complete the list of 4, or would you pick other ones entirely?

Well, if I had to pick four armies showing the spectrum of 40k, I'd take

Eldar (fast, fragile, huge fire power) subsuming DE and Necrons,
Tyranids (horde, cc oriented) subsuming Orks,
Space Marines (MEQ) subsuming all marine-like armies,
Daemons (elite, cc oriented).

The trick is to use allies in your armies and build diverse all round armies

Army 1: All of the imperium
Army 2: All Eldar + Tau
Army 3: Necrons
Army 4: Chaos + Orks

7 would be better Since chaos + orks and eldar + tau isn't ideal as is skipping tyranids.

If you want a cross section of tournament play just look at the tournament winners and make a cross section of the top 10 payers this should be easy.

I think the original Dawn Of War did it pretty well with Space Marines, Orks, Eldar and Chaos.

It's not a good cross section of play styles, but I would do:

Space Marines
Imperial Guard
Chaos Space Marines
Orks

This gets you the archetypical Space Marines in both good and bad flavors, which also allows you to broach the Horus Heresy and the larger storyline of the Imperium. Then you get the grunts. There's no way to present the Space Marines as superhuman if there's no humanity to contrast them against. Then, you have the orks, which are the alien race that is most distinctive to GW. Tyranids are fantastic foes, but very similar to many other alien gribblies, and the Tau, Eldar, and Dark Eldar just don't come across as the massive threats that the pervasive orks do. Necrons are pretty good, and a galaxy-wide threat, but don't seem as distinctive as Orks. Plus, if you have Orks and CSM, you have both 'elite' and 'grunt' enemies, and elite and grunt good guys.

Tau. The anti 40k faction.

Ork or Nids For the horde aspects and aliens.

Space Marines or Imperial Guard

Deamons or Chaos Space Marines

The issue here is that 40k steals borrows liberally from just about everything else out there. So your 40k could be very different from mine, which is very different from what the person you're introducing the game to would what to play.

The only constant is probably Space Marines, but that's because they're Games Workshop's most heavily advertised iconic models, and the ones that they have the widest selection of (i.e. more opportunities for you to buy stuff).

My family's campaign uses the following armies:

Tau Empire (secretly guided / manipulated by Tzeentch)

Daemons of Nurgle (the Tau think they're a "friendly" alien species)

Necrons (mechanical arthropods that rebelled, Matrix-style, against the terminator overlords)

Grey Knights (commanded by a female Inquisitor and sent to purge the Eastern Fringe of chaos)