Larry correctly got what B should pick if A picks HHH, to give B a 7:1 chance of winning.
While there are 8 different choices for how three tosses could show up, there are really only 4 when you consider symmetry.
HHH HHT HTH THH
And by switching heads and tails you will have the solutions for the rest:
TTT TTH THT HTT
The solution for HHT is similar to that for HHH, but gives B only a 3:1 chance of winning.
The remaining two are more difficult and give B only a 2:1 chance of winning.
When you get them all, you will discover that this contest is non-transitive, like rock-paper-scissors, where rock beats scissors beats paper beats rock beats...