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Chapter 1: Problem 14

How many 5 -card poker hands are there?

Short Answer

Expert verified
There are \(2,598,960\) different 5-card poker hands possible in a standard deck of 52 cards.

Step by step solution

01

Understanding combinations

Combinations are a way to count how many ways we can choose a specific number of items from a larger number of items without considering the order. The formula for combinations is given by: \(C(n, r) = \frac{n!}{r!(n-r)!}\) where \(C(n, r)\) represents the number of combinations of selecting r items from a set of n items, n! is the factorial of n, and r! is the factorial of r. In our case, n = 52 (total number of cards), and r = 5 (number of cards in a poker hand).
02

Calculating the factorials

First, let's calculate the factorials we need for our problem: \(52! = 52\times51\times50\times49\times48\times...\times3\times2\times1\) \(5! = 5\times4\times3\times2\times1\) \((52-5)! = 47! = 47\times46\times45\times...\times3\times2\times1\)
03

Applying the formula for combinations

Now, let's use these factorials in our combination formula: \(C(52, 5) = \frac{52!}{5!(47!)}\)
04

Simplifying the expression and finding the answer

Cancel the common terms between the numerator and the denominator: \(C(52, 5) = \frac{52\times51\times50\times49\times48}{5\times4\times3\times2\times1}\) Then, cancel out the common factors between the numerator and the denominator: \(C(52, 5) = \frac{2\times26\times17\times50\times49\times48}{2\times1\times3\times1\times1}\) \(C(52, 5) = 26\times17\times50\times49\times48\) Perform the numerical calculations: \(C(52, 5) = 2,598,960\)
05

Final Answer

There are 2,598,960 different 5-card poker hands possible in a standard deck of 52 cards.

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