11 politicians have formed a number of committees. every committee has exactly 6 members. the first ten politicians are in exactly 7 committees each. if the eleventh politician is in exactly n committees, what is the sum of al possible values of n?
Answers
Answer:
The first ten politicians being on 7 committees each means that they have 70 total seats on committees among them.
Minimum number of committees: Committees having exactly 6 members means that one committee would be 6 seats, two committees would be 12 seats, etc. So 11 committees with 66 seats wouldn’t be enough committees (since the first 10 have 70 seats), so we’d have to have AT LEAST 12 committees, meaning 72 seats.
Maximum number of committees: On the other hand, if we assume that politician #11 is on every committee, then that means the other 5 seats on each committee must be filled by the first ten politicians. Since the first ten politicians have 70 seats among them, that means we would have 70/5=14 committees, AT MOST.
So that means between 12 and 14 committees. That means #11 sits on 72–70=2 of 12 committees, 78–70=8 of 13 committees, or 84–70=14 of 14 committees. The sum of these possibilities is 2+8+14=24.