Question: A mathematically inclined club is forming a recruitment committee with five members. They have calculated that there are 8,568 ways to form this committee. Two of the club members are named Joaqu´in and Ana. The club calculates that 1,820 of the possible committees would have Joaqu´in on them but not Ana, 1,820 would have Ana but not Joaqu´in, and 560 would have both Joaqu´in and Ana.
(a) How many potential committees have either Joaqu´in or Ana?
(b) How many potential committees have neither Joaqu´in nor Ana?
(c) One semester, Joaqu´in and Ana carpool to meetings, so they insist that if either one of them is on the committee, then both should be on the committee. How many potential committees meet this condition?
(d) Later, Joaqu´in and Ana have a fight and refuse to work together. Joaqu´in says that if Ana is on the committee, he won't be on it, and Ana says that if Joaqu´in is on the committee, she won't be on it. How many of the potential committees meet this condition?