I'm going to be a bit blunt here, but you guys don't seem terribly good with math.
The T10 helm (iLvL 251) costs 1100 justice points.
A triumph badge converts into ~3 justice points. You need about 350-400 triumph badges to buy one piece of T10 gear.
That's (about) 80 random heroics. For one piece of gear. About 250-300 full random heroic runs for 4pc10.
At 15mins a piece, that's 70-80 hours of running random heroics.
All for gear that's going to be replaced with your first quest rewards and/or green drops.
Clearly, not a winning strategy.