Question

A hand of 5-card draw poker is a simple random sample from the standard deck of 52 cards. How many 5 draw poker hands are there? In 5-card stud poker, the cards are dealt sequentially and the order of appearance is important. How many 5-stud poker hands are there?

Answer 1

Answer:

There are 2,598,960 5 draw poker hands are there.

There are 311,875,200 5-stud poker hands.

Step-by-step explanation:

When the order is not important, we use the combinations formula:

For example, eating an apple and an orange is the same as eating an orange and an apple.

$C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

When the order is important, we use the permutations formula:

For example, a 3 digit credit card password, 123 is different than 213.

The number of possible permutations of x elements from a set of n elements is given by the following formula:

$P_{(n,x)} = \frac{n!}{(n-x)!)}$

A hand of 5-card draw poker is a simple random sample from the standard deck of 52 cards. How many 5 draw poker hands are there?

Here, the ordering is not important, so we use the combinations formula.

$C_{52,5} = \frac{52!}{5!47!} = 2598960$

There are 2,598,960 5 draw poker hands are there.

In 5-card stud poker, the cards are dealt sequentially and the order of appearance is important. How many 5-stud poker hands are there?

The ordering is important, so we use the combinations formula.

$P_{(52,5} = \frac{52!}{47!)} = 311875200$

There are 311,875,200 5-stud poker hands.