Is it permissible to combine the intention of making up missed fasts (qada'), expiation fasts (kaffarah), and the fast of Prophet David (peace be upon him), in the event that the number of days for the make-up and expiation fasts is unknown?

Fasting in fulfillment of a missed obligation (qada') and fasting as expiation (kaffara) are obligatory. Fasting one day and breaking fast one day is supererogatory (mustahabb). It is not permissible to combine the fasting of two obligatory fasts, nor to combine an obligatory fast with a supererogatory one. Rather, each fast must be observed on its own day. If you do not know the exact number of days of missed fasts you owe, then fast what you believe most likely clears your responsibility.