A standard deck of cards fifty-two cards has thirteen ranksR={A,2,3,4,5,6,7,8,9, T , J, Q, K}and four suitsS={,,,}. 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 isP. In terms of ranking, the professor outranks the king.

Question 1.Consider seven-card hands drawn from the MAT133 deck.

i) A flush occurs when all cards in a hand have the same suit; rank does not matter. For example, the seven-card handP, 3, 8,K,Q, 2, 7is a flush. How many seven-card hands drawn from the MAT133 deck are flushes?

ii) A straight occurs when the seven cards have consecutive ranks. Note that theAcan be used to finish a straight. For example, the seven-card hand 9,T,J,Q,K,P,Ais a straight, but the handQ,K,P,A, 2, 3, 4is 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 first set of four cards). For example, the seven-card hand 3, 3, 3, 3,A,A,Ais 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 flush is not a straight, and a straight flush is not a flush. To avoid complication, for the purposes of this question, a straight flush is both a straight and a flush.

Question 2.

i) Suppose you are dealt a seven-card hand from the MAT133 deck. How many of those hands include exactly one professor and exactly one smile?

ii) Suppose you are dealt a seven-card hand from the MAT133 deck. How many of those hands include at least one professor and at least one smile?