A standard deck of cards 52 cards has 13 ranks R={A,2,3,4,5,6,7,8,9,10, J, Q, K}and four suits S={♥,♦,♣,♠}. Now, we play cards with an extended deck. We have an additional suit of cards: the dollar suit $ ; and an additional rank: Price . For example, the “Price of dollar” card is P$ Q3. 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♥, 4♠, 4♦, P $ , P ♥, 2? $} is valid. Examples of invalid hands are {J ♠, J $ ?, J ♥, Q$, Q♣, 7♥}, {5♥,5♠,5?$,P$? ,P♦,4♣}, and {P$?,P♦,P♠,2?$,2♦,A♣}. ,5♠,5?,P?,P♦,4♣}, and {P?,P♦,P♠,2?,2♦,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.
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
A standard deck of cards 52 cards has 13 ranks R={A,2,3,4,5,6,7,8,9,10, J, Q, K}and four suits S={♥,♦,♣,♠}. Now, we play cards with an extended deck. We have an additional suit of cards: the dollar suit $ ; and an additional rank: Price . For example, the “Price of dollar” card is P$
Q3.
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♥, 4♠, 4♦, P $ , P ♥, 2? $} is valid. Examples of invalid hands are {J ♠, J $ ?, J ♥, Q$, Q♣, 7♥}, {5♥,5♠,5?$,P$? ,P♦,4♣}, and {P$?,P♦,P♠,2?$,2♦,A♣}.
,5♠,5?,P?,P♦,4♣}, and {P?,P♦,P♠,2?,2♦,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.
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