The first weapon that you win will be appropriate for your loot-specialization. If you are fortunate enough to win one of these, you will get a weapon relevant to you. (As an example, a hunter's first heirloom WILL be Hellscream's Warbow.) Even if you have a better weapon already (such as a Heroic when a Normal item dropped), I recommend keeping the heirloom, as it will outlast your current item as you level up. Not to mention, they can be used for transmogrification!
https://www.wowhead.com/npc=71865/garrosh-hellscream#comments
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I recall a blue post answering this last question. The for-your-spec guarantee is separate for every difficulty, so you don't need to avoid killing flex garrosh until you get a normal heirloom.
Только первая фамилька при первом килле должна падать на твой класс/спек. Остальные - рандом