Terraria – Is it possible to have the Corruption and the Crimson Biomes in the same world

Crimson and corruption are mutually exclusive - only one can exist in a world without the player bringing in items from another world. Breaking Demon Hearts in Crimson chasms will achieve a comparable effect (the boss is different, prizes are very similar, meteors do fall) to breaking shadow orbs in Corruption.

Although this is not the most accessible way to do it, the easiest way is to use the Clentaminator with Green Solution.

The more accessible ways are to use the Purification Powder from a Dryad, bomb the Crimson with anything that breaks blocks when it explodes or mine it with a pickaxe (power 65% or higher, or in terms of tools: nightmare/deathbringer pickaxe or better).