Let G be a group of order n that acts operates non-trivially on a set of size r. Prove that if n > r!, then G has a proper normal subgroup. (A proper subgroup of G is a subgroup that is neither trivial nor equal to G.)

The host of a game show is holding a bag with chips to see what prizes the contestants will play for. five (5) of the chips say “new car” ten (10) of the chips say “new tv” three (3) of the chips say “trip to france” contestant named “patty” reaches into the bag. a. what is the probability of choosing a tv chip? b. what is the probability of choosing a car chip next without replacing the 1st chip? c. are these dependent or independent events?