Required and forbidden members

For an unordered group, required members already occupy places; forbidden members are removed from the available pool. All remaining choices use combinations. If two named people must belong to a four-person committee, only two places remain. The required pair is fixed, so it contributes one choice, not an extra factorial. Start with the committee size, subtract required members, remove all required and forbidden members from the pool, and select the remaining places. For exact category quotas multiply selections from disjoint categories. Example: from nine people, a four-person committee must include Ana and exclude Ben. The remaining three members are chosen from seven people: 73=35. Check: “exactly one of Ana and Ben” gives two disjoint membership cases. “At least one” also allows both; do not confuse the two.