A standard deck of cards fty-two cards has thirteen ranks R = {A, 2, 3,4, 5, 6, 7,8, 5LT, J, Q, X} and four suits S = {C9, ,,.}. In MAT 133, we play cards with an extended deck. We have an additional suit of cards: the smile suit ; and an additional rank: the Professor. For example, the \"professor of smiles\" card is PCS. In terms of ranking, the professor outranks the king. Question 1. Consider seven-card hands drawn from the MAT133 deck. i} A ush occurs when all cards in a hand have the same suit; rank does not matter. For example, the seven-card hand P0, 36', 80, KO, Q0, 20, 7'? is a ush. How many seven-card hands drawn from the MAT133 deck are ushes? ii} A straight occurs when the seven cards have consecutive ranks. Note that the A can he used to nish a straight. For example, the seven-card hand 90, T', JO, Q, KO, P69, A' is a straight, but the hand QU', K., P', AC9, 20, 3., 4C? is not a straight. How many seven-card hands drawn from the MAT133 deck are straights? iii} An overfull house occurs when the seven-card hand consists of four cards of the same rank, together with three cards of the same rank [which must be different from the rst set of four cards}. For example, the seven-card hand 3., 30, 3G), 3', A0, AG), A0 is an overfull house. How many seven-card hands drawn from the MAT133 deck are overfull houses? Note: For those of you who know about card hands, we usually say that a straight ush is not a straight, and a straight ush is not a flush. To avoid complication, for the purposes of this question, a straight ush is both a straight and a ush