Math Lesson #11: How many hands of 5 cards?

Well, there are 52 cards in a deck, so there are 52 choices for the first card:

Leaving 51 choices for the second card:

And 50 choices for the third card:

And 49 for the fourth card:

And 48 for the fifth card:

So, this is 52 x 51 x 50 x 49 x 48 = 311,875,200

But, in poker it doesn't matter if you pull an Ace first or last... so we need to ignore the different order our cards can land in our hand. We saw in Math Lesson #10 that there are 5! = 120 ways of ordering 5 cards, so we have to divide our 311,875,200 by 5!...

311,875,200 / 120 = 2,598,960

So, there are almost 2.6 million unique poker hands...

Note, we can write (52 x 51 x 50 x 49 x 48) as 52! / 47!.... the division cancels out all the lower numbers

We can even go further and rewrite it as 52! / (52-5)!

Then the whole thing, also taking care to cancel out the different orders is: 52! / ((52-5)! x 5!)