COUNTIFS formula to exclude respondents - help, please!

Hello, I've looked through the forum and I think I've found the answer but I would love input from those who know Excel. I am trying to determine the true number of people who have employee, goverment (public), or no insurance; the categories should be mutually exclusive but some people responded 'yes' (1, whereas no is 2) to multiple categories. For example, respondents B and D answered correctly in the table below. If they answered yes (1) to Employee, that would supersede any other answer (such as respondent C). If I need to find out how many people answered (1) to employee, whether alone or in combination with some other answer, what would be the best formula? (And could I use that same formula to find out how many people just answered yes to 'public' or 'no'?)

Thank you so much!