Explanation 1) Definition: The Fundamental Counting Principle (FCP): If we want to perform a series of tasks and the first task can be done in a ways, the second can be done in b ways, the third can be done in c ways, and so on, then all the tasks can be done in a × b × c … ways. 2) Calculation: Given, We assume that we are drawing five cards from a standard 52-card deck that contain two kings and three aces For find that number of ways, we follow the below two stages. Stage 1: Select two kings in the deck (i.e.) we must select two suit kings For we get the number of ways with combination C ( n , r ) ⋅ C ( 13 , 1 ) ⋅ C ( 4 , 2 ) Stage 2: Select three aces in the deck. (i.e.) we must select three suit aces. For we get the number of ways with combination C ( n , r ) ⋅ C ( 13 , 1 ) ⋅ C ( 4 , 2 ) Thus, we can apply the fundamental counting principle. Stage 1: C ( 13 , 1 ) ⋅ C ( 4 , 2 ) First we find that C ( 13 , 1 ) Using the formula for combinations, C ( n , r ) = n ! r ! ( n − r ) ! Here, n = 13 and r = 1 C ( 13 , 1 ) = 13 ! 1 ! ( 13 − 1 ) ! C ( 13 , 1 ) = 13 ! 1 ! ( 12 ) ! C ( 13 , 1 ) = 13 ⋅ 12 ! 1 ⋅ 12 ! C ( 13 , 1 ) = 13 Next we find that C ( 4 , 2 ) : Using the formula for combination, C ( n , r ) = n ! r ! ( n − r ) ! C ( 4 , 2 ) = 4 ! 2 ! ( 4 − 2 ) ! C ( 4 , 2 ) = 4 ! 2 ! 2 ! C ( 4 , 2 ) = 4 ⋅ 3 ⋅ 2 ! 2 ! 2 !

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ISBN: 8220103632027

Author: Pirnot

Publisher: Pearson Education (US)

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Chapter 12.3, Problem 80E

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The number of ways can we construct a five hand that contains two kings and three aces.

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Chapter 12.3, Problem 79E

Chapter 12.3, Problem 81E

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The number of ways can we construct a five hand that contains two kings and three aces.

Solution Summary: The author explains the Fundamental Counting Principle (FCP): if we want to perform a series of tasks, we must select two kings in the deck.

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The number of ways can we construct a five hand that contains two kings and three aces.

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