Am I completely misunderstanding the problem, or is the math in the example solution wrong?
Bursting 1 first gives you 3×1×5=15 coins. <-- OK...
Bursting 3 first gives you 1×3×1=1 coins. (using left virtual balloon) <-- Should be 3?
A possibility would be bursting 1, then 5, then 3, which gives you a total of 3×1×5+3×5×1+1×3×1=33 coins. <-- Should be 3x1x5+1x5x1+1x3x1=24?
The first error seems to be a transcription error, it's correct in their graph representation further down.
The second one isn't an error, but a poor explanation. After a balloon is burst, the balloons now have new neighbors. That is, it isn't this:
It's: The distance between the balloons doesn't make them not neighbors.If you have "3 1 5" and you burst 1, you gain 3x1x5 points and the state becomes "3 5", with the two remaining balloons being adjacent to each other.
The "1x3x1=1" part for the earlier example is a typo indeed, it should be 3.