(1) Given a regular octagon, in how many ways can we color one diagonal red and another diagonal blue so that the two colored diagonals cross (in the interior)? Consider rotations and reflections distinct.
(2) There are 2 senators from each of the 50 states. We wish to make a 3-senator committee in which no two of the members are from the same state. In how many ways can we do it?
(3) 10 chairs are arranged evenly around King Arthur's round table. In how many ways can 10 knights be seated in the chairs if Sir Lancelot and Sir Gawain insist on being seated diametrically opposite from each other AND Sir Galahad & Sir Percival also need to be seated diametrically opposite.
(4) In triangle ABC, we have \(\angle B=60^\circ\), \(\angle C=90^\circ\), and AB = 2. Let P be a point chosen uniformly at random inside ABC. Extend ray BP to hit side AC at D. What is the probability that \(BD<\sqrt 2\)?
(1) There are 6*16 = 96 ways of choosing the diagaonals.
(2) There are 422 ways of choosing the senators.
(3) There are 8*9*8 = 288 ways of seating the knights.
(4) The probability is 3/5.
Hope this helps!