Is it obligatory for someone who embraced Islam on the twentieth of last Ramadan, and was ill and unable to fast, to make up the last ten days or the entire month?

The person who embraced Islam in Ramadan is addressed with the obligation to fast the remainder of the month, and to make up for it (qada') if able, or to pay expiation (kaffara) if permanently unable. As for the days of the month that passed before his conversion to Islam, he is not obligated to make them up. He should make up the day he embraced Islam as a precautionary measure, due to a difference of opinion among scholars on this issue.