Defining a counting complement

Choose the outcome universe first. “At least one” is the complement of “none”; “at most one” permits zero or one. Counts may be added only for mutually exclusive cases. A committee either meets the condition or lies in its complement, never both. Subtract unwanted committees from all committees of the same size and eligible pool. For a quota, list allowed category counts and check that enough objects exist in each category. Add their combination-only counts, or subtract the shorter list of forbidden cases. From four painters and five dancers, three-person groups with at least two painters number 42 51+ 43=30+4=34. “At most one painter” is the complementary count 93-34=50. Check: “not both Ana and Ben” excludes committees containing both, but permits neither or exactly one. Its complement is not “neither”.