The final pressure is 420 Torr.

When we connect the bulbs, the total volume #V_"tot"# of the system is

#V_"tot"# = (1.0 + 1.0 + 0.5) L = 2.5 L

No we can use Boyle's Law to calculate the partial pressure of each gas after it has expanded into the new volume.

For N₂: #P_2 = P_1 × V_1/V_2 = 635"Torr" × (1.0"L")/(2.5"L")# = 254 Torr (2 significant figures)

For Ne: #P_2 = P_1 × V_1/V_2 = 212"Torr" × (1.0"L")/(2.5"L")# = 84.8 Torr (2 significant figures)

For H₂: #P_2 = P_1 × V_1/V_2 = 418"Torr" × (0.5"L")/(2.5"L")# = 83.6 Torr (1 significant figure)

Each gas exerts it pressure independently of the others, so we can apply Dalton's Law of Partial Pressures.

#P_"tot" = P_"N₂" + P_"Ne" + P_"H₂"# = (254 + 84.8 + 83.6) Torr = "422.4"Torr = 420 Torr (2 significant figures)