Explanation Given Information: Poker hand of four of a kind is all four of one denomination, one of another. Formula repeated: The Decision Algorithm is to make a set of decisions to determine the number of possible outcomes. The Multiplication Principle is, If there is a sequence of choice with r disjoint steps and step 1 has n 1 possible outcomes, step 2 has n 2 possible outcomes and so on and step r has n r possible outcomes with no two outcomes same, then the total number of outcomes are n 1 × n 2 × ⋯ × n r . The value of combination of n items taken r at a time is, C ( n , r ) = n × ( n − 1 ) × ⋯ × ( n − r + 1 ) r ! Calculation: Consider poker hand of four of a kind is all four of one denomination, one of another. Use the Decision Algorithm to make a set of decisions to determine the number of possible outcomes. Consider the Decision Algorithm, there are 5 empty slots and choose all four of one denomination, one of another. Step 1: Choose 1 denomination for 4 cards. Since, there are 13 denominations and 1 to choose. Thus, Step 1 has C ( 13 , 1 ) choices. Step 2: Choose 4 cards of the above denominations. Since, there are 4 suits and 4 cards to choose. Thus, Step 2 has C ( 4 , 4 ) choices. Step 3: Choose a denominations for the single card. Since, there are 12 cards of other denominations and 1 to choose. Thus, Step 3 has C ( 12 , 1 ) choices. Step 4: Choose 1 card of the denomination above. Since, there are 4 suits and 1 to choose. Thus, Step 4 has C ( 4 , 1 ) choices. Use the Multiplication Principle as below, Since, there are 4 steps, r = 4 and no two are same. Thus, the total number of outcomes is, n 1 × n 2 × n 3 × n 4 = C ( 13 , 1 ) × C ( 4 , 4 ) × C ( 12 , 1 ) × C ( 4 , 1 ) Now, the value of C ( 13 , 1 )

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Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 6.4, Problem 58E

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To calculate: The number of poker hand of four of a kind which is all four of one denomination, one of another where a poker hand consists of 5 cards from a deck of 52 cards.

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Chapter 6.4, Problem 57E

Chapter 6.4, Problem 59E

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Chapter 6 Solutions

Finite Mathematics

Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - Prob. 8E Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E

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The number of poker hand of four of a kind which is all four of one denomination, one of another where a poker hand consists of 5 cards from a deck of 52 cards.

Solution Summary: The author explains how the Decision Algorithm calculates the number of possible outcomes for poker hand of four of a kind.

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To calculate: The number of poker hand of four of a kind which is all four of one denomination, one of another where a poker hand consists of 5 cards from a deck of 52 cards.

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