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I calculate it as seven 1 from Barbarian level 4 and then 6 from 16 levels in Fighter "When you reach 4th level, and again at 6th, 8th, 12th, 14th, 16th, and 19th level"

Someone keeps telling me it's six only because I lose the 19th level increase, but isn't that cancelled out by gaining it from the 4th Barbarian level?

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You are correct, you get 7 ASIs

You get 1 at Barbarian level 4.

You get 1 at each of Fighter levels 4, 6, 8, 12, 14 and 16

For a total of 7

Ability score increases are computed for each class separately, and they go by class level and not character level.

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You are correct, a Barbarian 4/Fighter 16 gains 7 ability score increases over the course of their career. 6 from their Fighter levels and 1 from their Barbarian levels.

Obviously, if you had gone, for example, Barbarian 2/Fighter 18, you would have "lost" the level 19 ability score increase as was suggested to you, but since you took enough levels to gain one from Barbarian, it balances out.

Purely as a side note, the reason why the final ability score increase is at level 19 instead of level 20 (which would maintain the pattern of multiples of 4) is probably so that you can take 1-level dips in other classes without losing an ability score increase.

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    \$\begingroup\$ I'm pretty sure the reason for the final ASI being at level 19 was so they could put the class capstone at 20, while still keeping with their general design rule of not giving ASI's and major features at the same level. Being able to take a 1 level dip in place of some of the more lacklustre capstones is just a nice bonus ;) \$\endgroup\$ – Andrew Aug 26 '15 at 13:55
  • \$\begingroup\$ I agree with @Andrew -- there's nothing to suggest they wanted to especially facilitate multiclassing (and much to suggest that they don't), particularly not dipping "just" for some mechanical benefit or synergy. (Not that it really matters why it's set up that way.) \$\endgroup\$ – PurpleVermont Aug 26 '15 at 15:13

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