Please provide a correct and detailed explanation for Question 3 A standard deck of cards fifty-two cards has thirteen ranks R = {A, 2,3,4, 5, 6, 7, 8, 9, J, Q, K} and four suitsS = {♡, 0, 4, A}. In MAT 133, we play cards with an extended deck. We have an additional suit of cards: the smilesuit ©; and an additional rank: the Professor. For example, the "professor of smiles" card is PO.Question 1 Write the following sets:1. S' the suits of the MAT 133 deck2. R' the ranks of the MAT 133 deck3. Which cards are in the MAT 133 that are not in the standard deck?(a) Write your answer as a set using R, S, R', S' and the set operations: × \ U { }.(b) List the elements which are in the MAT 133 deck, but not the standard deck.4. How many cards are there in the MAT 133 deck which are not in the standard deck?Question 2 Consider five-card hands drawn from the MAT 133 deck.1. How many hands contain at least one Professor? For example, {34, 5©, PO, K♡,7%} is such a hand.2. A full house is a five-card hand with a three-of-a-kind and a pair of different ranks. How many full houses canbe formed using the MAT133 deck?Question 3 A fuller house is a six-card hand with a three-of-a-kind, a pair, and a singleton card of different ranks.How many fuller houses can be formed using the MAT133 deck if none of the ranks can be consecutive?For example, {4♡, 44, 40, P©, PV, 2©} is valid. Examples of invalid hands are {JA, J©, J♡,Q©,Q4, 70},{50, 54, 5©, PO, PO, 4%}, and {P©,PO,P&,2©,20, A}.Assume the ranks, in increasing order, are A, 2,3, 4, 5, 6, 7, 8, 9, T, J,Q, K, P, and that P and A are consideredconsecutive ranks.

Name: Please provide a correct and detailed explanation for Question 3 A sta
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Description: Please provide a correct and detailed explanation for Question 3 A standard deck of cards fifty-two cards has thirteen ranks R = {A, 2,3,4, 5, 6, 7, 8, 9, J, Q, K} and four suitsS = {♡, 0, 4, A}. In MAT 133, we play cards with an extended deck. We ha

Given information: In MAT 133, the ranks is R = {A, 2, 3, 4, 5, 6, 7, 8, 9, J, Q, K, P}. The number…

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Question 3 A fuller house is a six-card hand with a three-of-a-kind, a pair, and a singleton card of different ranks. How many fuller houses can be formed using the MAT133 deck if none of the ranks can be consecutive? For example, {4♡, 44, 40, P©, P♡, 2©} is valid. Examples of invalid hands are {JA, JO, JV, QO, Q4, 70}, {50, 54, 5©, PO, PO, 4%}, and {P©,P\,P&,2©,20, A$}. Assume the ranks, in increasing order, are A, 2, 3, 4, 5, 6, 7, 8, 9, T, J, Q, K, P, and that P and A are considered consecutive ranks.