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Poker Hands Using combinations, calculate the number of each type of poker hand in a deck of cards. (A poker hand consists of 5 cards dealt in any order.)$$\begin{array}{l}{\text { a. Royal flush }} \\ {\text { b. Straight flush (not including a royal flush) }} \\ {\text { c. Four of a kind }} \\ {\text { d. Full house }}\end{array}$$

Intro Stats / AP Statistics

Chapter 4

Probability and Counting Rules

Section 4

Counting Rules

Sampling and Data

Probability Topics

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University of North Carolina at Chapel Hill

Oregon State University

Lectures

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for this question. We want to determine the number off each type off poker hand. Remember that there are five guards in a poker hand that they're a royal flush. Be straight flush, excluding the royal flush C four for kind, The full house. Let's look at the first poker hand. Royal flush. So this hand consists off. Okay, then. Jack. Queen, King Ace. All in the same suit. I know that the ease here is ranked higher than the king. So this hand is the term in completely by this soup. Recall that there are four suits in a standard deck the heart's Thailand clubs and speeds. So there would be for royal flush hands because there are four suits in a standard IQ that they're being Streep flush. So a straight flush is five guards in a sequence long in this same suit. So let's list all the ranks off the cards. Okay, So there are 13 ranks ranging from a space up to King note that in straight flush and is is ranked below a two. So on example off a straight flush hand would be made up off the cards A's to three, four and five another straight flush hand would be made up off the guards, too. Three, 45 and six notice that the hand is determined by the No West Guard and the lowest ranked card. So let's look at the possible lowest cards in a straight flush. That would be he's to 34567 eight in nine. That's a total of nine straight flush hands in a particular suit. And there are four suits, so there would be nine times for 36 different straight flush hands. Not the nine. Here is the number of straight flush hands in a soup forest. The number off suits, not her see four of a kind. So this hand has four guards off the same rank point. So let's look at the particular rank. So let's say we have form jacks so that the fifth card I would just be some other card. Remember, a hand. A poker hand has five guards, so there are combination for taken four ways off getting the four jacks, and there would be a combination 48 taken. One wastes off, choosing the other card. We use 48 because a standard deck has 52 cards and four of them the four jacks had already been used, So we can only choose the one undercard from the 48 remaining cards. We multiply this by the fundamental accounting rules. No, there are 13 possible ranks to choose from, so the total number of ways that we can have a four off a kind hand would be 13 times combination for taking for times combination. 48 take in one. Right, so 13. Here's for dough number off ranks, and this would be equal toe 13 times, one times 48. That's equal 624. Four of a kind hence so that 30 full house a full house. It should three off a kind end a pair. So let's look at the particular full house hand. So, let's say, are three or four kind are aces and a pair are objects. There are combination for taken three ways of getting three aces, and there are a combination for taken two ways of getting a pair of jacks. By the fundamental counting rule. We multiply those two and they said the number off ways you can have a full house with three aces and jacks Now there are permutation 13 taken to ways of choosing two ranks for our full house and note that order is important here. So the total number off full house hands would be permutation. 13. Take intuited time times combination for taken three times combination for taken to no, that Ah, full house with three aces and two jacks would be different from, Ah, full house with three jacks and two aces. So order. It's important. That's why we're using permutation. This would be equal to 1 56 times four I'm six, which is equal to 3744 different full house hands.

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