Do I need to buy Cataclysm to unlock the new race/class combinations in World of Warcraft?

Solution 1:

Like the content within the most recent 4.0.3 patch (i.e., the shattering), all the changes and new quests, etc. are available to anyone with the base game.

According to the official patch notes: http://us.battle.net/wow/en/blog/1214741

The new race / class combinations are already available in the game. (See the 2nd bulletpoint, reprinted below):

New Race/Class Combinations In the wake of a world on the brink of destruction, members of the Horde and Alliance have taken to new cultures and studies, mastering crafts previously foreign to them. Many existing playable races now have new class combinations. Check the official World of Warcraft: Cataclysm website for a breakdown of the new combinations.