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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 from a standard 52-card deck, calculated using the combination formula C(52, 5) = \(\frac{52!}{5!(52-5)!}\).

Step by step solution

01

Identify the number of distinct items

In this case, we have a standard 52-card deck with 52 distinct items (cards).
02

Identify the number of items to choose

We are asked to find the number of 5-card poker hands, so we need to choose 5 items (cards) from the 52 distinct items.
03

Apply the combination formula

The combination formula is given by: C(n, k) = \(\frac{n!}{k!(n-k)!}\) where n is the total number of distinct items, k is the number of items we want to choose, and ! denotes the factorial function. In our case, n = 52 and k = 5. Applying the combination formula: C(52, 5) = \(\frac{52!}{5!(52-5)!}\)
04

Simplify the expression

Now we will simplify the expression: C(52, 5) = \(\frac{52!}{5!(52-5)!}\) = \(\frac{52!}{5!47!}\) We can simplify this further using the property of factorials: \(a! = a 脳 (a-1) 脳 (a-2) 脳 ... 脳 1\) Using this property, we can expand 52! as follows: C(52, 5) = \(\frac{52 脳 51 脳 50 脳 49 脳 48 脳 47!}{5!47!}\) Notice that we can now cancel out the 47! terms: C(52, 5) = \(\frac{52 脳 51 脳 50 脳 49 脳 48}{5!}\) Now we expand 5!: C(52, 5) = \(\frac{52 脳 51 脳 50 脳 49 脳 48}{5 脳 4 脳 3 脳 2 脳 1}\) Finally, we can simplify this expression by performing the arithmetic operations: C(52, 5) = 2,598,960
05

Interpret the result

We found that there are 2,598,960 different 5-card poker hands possible from a standard 52-card deck.

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