How can one make up for accumulated fasting days for Ramadan and for oaths, when the number of days for each is indistinguishable, and is expiation due for the missed Ramadan days from 2009 and 2010?

If the days to be fasted are intended as an expiation for an oath, then expiation is not initially by fasting unless one is unable to feed ten needy persons, clothe them, or free a slave, as Allah Almighty says: "Its expiation is the feeding of ten needy persons from the average of that which you feed your own families, or clothing them or freeing a slave. But whoever finds not, then a fast of three days." (Al-Ma'idah: 89).

However, if these are days you swore to fast, then it is preferable to fast them, or to not fast them and pay an expiation for an oath. The intention for obligatory fasting must be made at night, and the type of obligation (make-up or expiation) must be specified. If the reason for the obligation is unknown, intending an obligatory fast is sufficient due to necessity.

Therefore, fast what you are certain is a make-up fast with the intention of making up, and what requires expiation with the intention of expiation, and what its reason is unknown with the intention of an obligatory fast. If you delayed making up Ramadan until another Ramadan began without an excuse, then you must pay a fidyah (expiation) of feeding one needy person for each day, unless you were unaware of the prohibition of delay, in which case nothing is required of you.