It takes $4000 of tuition and other qualified educational expenses (QEE) (e.g. books) to get the maximum American Opportunity credit (AOC). That's you goal: free up as much QEE as you can to claim the credit, which you can't do when the scholarship is allocated to QEE. By allocating the scholarship to room & board (R&B) a non qualified expense, you free up QEE for the AOC. You allocate all you can, up to $4000.
So, in your case, you allocate all $2740 to R&B. That allows you to claim the $1665 QEE (1415 + 250 = $1665) for the AOC. The AOC is 100% of the first $2000 QEE and 25% of the next $2000. You will get a $1665 credit.
Your student will have to report $2740 of taxable scholarship on his tax return. If he has no other income, he will pay no actual income tax, because his total income is less than the $14,600 filing requirement.
