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Looking at the ability score table I can go down to 7 and get 4 points into my pool, so let us say I have 15 points to spend and dropping strength to 7 I have 19 points according to my understanding. But let us say I choose human and put my 2 ability points (after the point buying into strength and the minimum ability score set by the DM would be 7) I could go down to 5 at the beginning. Looking at the sequence 10:0; 9:-1; 8:-2; 7:-4; 6:-6; 5:-8 I would get 8 points into my pool. Is that possible? Do exit any rules for that since I have not found or overlooked it?

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It is not allowed by the rules.

Your DM could allow it, of course, but that's a separate matter.

You can't buy an ability score lower than 7 for the same reason you can't buy an ability score higher than 18: it's not an option. The table only has costs for values from 7 to 18. We can't even know for certain what ability scores outside that range would cost, since the changes in cost don't follow an easy-to-derive pattern.

Furthermore, it's something that shouldn't be allowed. It encourages players to over-optimize their characters, making too-dumb-to-breathe musclebound brutes and too-ugly-to-look-at spellcasting geniuses. You end up with characters that are stronger than the game is balanced for in a few areas and utterly useless in others. Any DM who allows characters to purchase scores outside the 7-18 range deserves the dysfunctional party they'll end up with.

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    \$\begingroup\$ "Any DM who allows characters to purchase scores outside the 7-18 range deserves the dysfunctional party they'll end up with" is the kind of straight talk that needs to be heard/seen more often. \$\endgroup\$ Commented Apr 12, 2016 at 3:30

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