What is incumbent upon one who broke her fast on days during Ramadan—some with a valid excuse and some without—and did not make them up, and then forgot their number? And is it permissible for her to give charity instead of fasting?

You must make up for the days you missed fasting and feed a poor person for each day as expiation for the delay, as long as you are able to fast. Your obligation will not be fulfilled except by doing so, because feeding is not sufficient except for those who are unable to fast. As for determining the number of days, it is done by investigation and diligent effort.