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The goal is to determine the number of different five-card poker hands that consists of two pairs. In other words, we need to find the number of ways to pick 2 cards of one denomination, 2 cards of another denomination, and a final card that belongs to neither of the first two denominations. Our decision algorithm is a sequence of four steps. Step 1: Select 2 denominations for the pairs. Step 2: Select 2 cards from one selected denomination. Step 3: Select 2 cards from the other selected denomination. Step 4: Select 1 card that belongs to neither of the previously selected denominations. We begin with Step 1. There are 13 possible denominations and we will pick 2 of them. Because the order that we make this selection does not matter, we will use combinations to find the number of ways to pick 2 denominations out of 13. Therefore, in each case we will be using combinations to count the number of ways to take n items taken r at a time, which is calculated using the following formula. n! C(n, r) = r!(n - r)! Here we have n = 13 andr= 2. Substitute these values into the formula and simplify. n! C(n, r) = r!(n r)! C(13, 2) = 2!(13 – 2)! In other words, there are possible ways to pick 2 of the 13 denominations for the pairs.

The goal is to determine the number of different five-card poker hands that consists of two pairs. In other words, we need to find the number of ways to pick 2 cards of one denomination, 2 cards of another denomination, and a final card that belongs to neither of the first two denominations. Our decision algorithm is a sequence of four steps. Step 1: Select 2 denominations for the pairs. Step 2: Select 2 cards from one selected denomination. Step 3: Select 2 cards from the other selected denomination. Step 4: Select 1 card that belongs to neither of the previously selected denominations. We begin with Step 1. There are 13 possible denominations and we will pick 2 of them. Because the order that we make this selection does not matter, we will use combinations to find the number of ways to pick 2 denominations out of 13. Therefore, in each case we will be using combinations to count the number of ways to take n items taken r at a time, which is calculated using the following formula. n! C(n, r) = r!(n - r)! Here we have n = 13 andr= 2. Substitute these values into the formula and simplify. n! C(n, r) = r!(n r)! C(13, 2) = 2!(13 – 2)! In other words, there are possible ways to pick 2 of the 13 denominations for the pairs.

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter14: Counting And Probability
Section14.1: Counting
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The goal is to determine the number of different five-card poker hands that consists of two pairs. In other words, we need to find the number of ways to pick 2 cards of one denomination, 2 cards of another denomination, and a final card that belongs to neither of the first two denominations. Our decision algorithm is a sequence of four steps. Step 1: Select 2 denominations for the pairs. Step 2: Select 2 cards from one selected denomination. Step 3: Select 2 cards from the other selected denomination. Step 4: Select 1 card that belongs to neither of the previously selected denominations. We begin with Step 1. There are 13 possible denominations and we will pick 2 of them. Because the order that we make this selection does not matter, we will use combinations to find the number of ways to pick 2 denominations out of 13. Therefore, in each case we will be using combinations to count the number of ways to take n items taken r at a time, which is calculated using the following formula. n! C(n, r) = r!(n - r)! Here we have n = 13 andr= 2. Substitute these values into the formula and simplify. n! C(n, r) = r!(n r)! C(13, 2) = 2!(13 – 2)! In other words, there are possible ways to pick 2 of the 13 denominations for the pairs.
Transcribed Image Text:The goal is to determine the number of different five-card poker hands that consists of two pairs. In other words, we need to find the number of ways to pick 2 cards of one denomination, 2 cards of another denomination, and a final card that belongs to neither of the first two denominations. Our decision algorithm is a sequence of four steps. Step 1: Select 2 denominations for the pairs. Step 2: Select 2 cards from one selected denomination. Step 3: Select 2 cards from the other selected denomination. Step 4: Select 1 card that belongs to neither of the previously selected denominations. We begin with Step 1. There are 13 possible denominations and we will pick 2 of them. Because the order that we make this selection does not matter, we will use combinations to find the number of ways to pick 2 denominations out of 13. Therefore, in each case we will be using combinations to count the number of ways to take n items taken r at a time, which is calculated using the following formula. n! C(n, r) = r!(n - r)! Here we have n = 13 andr= 2. Substitute these values into the formula and simplify. n! C(n, r) = r!(n r)! C(13, 2) = 2!(13 – 2)! In other words, there are possible ways to pick 2 of the 13 denominations for the pairs.
Poker Hands A poker hand consists of 5 cards from a standard deck of 52. Find the number of different poker hands of the specified type. two pairs (two of one denomination, two of another denomination, and one of a third) For those unfamiliar with playing cards, here is a short description. A standard deck consists of 52 playing cards. Each card is in one of 13 denominations: ace (A), 2, 3, 4, 5, 6, 7, 8, 9, 10, jack (J), queen (Q), and king (K), and in one of four suits: hearts (), diamonds (+), clubs (), and spades (4). Thus, for instance, the jack of spades, Je, refers to the denomination of jack in the suit of spades. The entire deck of cards is as shown below. AV 6V 7V 89 10V JV Q♥ K♥ J• Q• 10 JA Q KA JA Q4 A 2+ 3+ 50 64 7 8 6 74 10+ K 84 94 A4 24 34 44 54 64 74 84 94 104 KA Step 1
Transcribed Image Text:Poker Hands A poker hand consists of 5 cards from a standard deck of 52. Find the number of different poker hands of the specified type. two pairs (two of one denomination, two of another denomination, and one of a third) For those unfamiliar with playing cards, here is a short description. A standard deck consists of 52 playing cards. Each card is in one of 13 denominations: ace (A), 2, 3, 4, 5, 6, 7, 8, 9, 10, jack (J), queen (Q), and king (K), and in one of four suits: hearts (), diamonds (+), clubs (), and spades (4). Thus, for instance, the jack of spades, Je, refers to the denomination of jack in the suit of spades. The entire deck of cards is as shown below. AV 6V 7V 89 10V JV Q♥ K♥ J• Q• 10 JA Q KA JA Q4 A 2+ 3+ 50 64 7 8 6 74 10+ K 84 94 A4 24 34 44 54 64 74 84 94 104 KA Step 1
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