I believe you are not correct. I'm sure there are private scholarships that do not allow the funds to be used for room board, but the article indicates that Pell grants can be applied differently than the way applied by the institution. Please see the following link
https://www.irs.gov/pub/irs-utl/Pell%20AOTC%204%20pager.pdf
What is the dependents date of birth? If it is 01-January the dependent is deemed to be a year older than the arithmetic indicates. This came to light when the credit was first implemented.