# What are Combinations?

A Combination is a type of calculation that tells us how many ways we can gather together a set of things without regard to the order in gathering those things.

#### Explanation:

A Combination is a type of calculation that tells us how many ways we can gather together a set of things without regard to the order in gathering those things.

A common question in combinations is to calculate how many poker hands are possible (the number of ways we can gather together a certain type of cards, such as a Royal Flush or a Pair or anything in between). We only care about the end result - the Royal Flush - and don't care that we were dealt the Ace first, then the 10, then the Queen, etc... If we did care, that would be a Permutation question.

The general formula for working with combinations is:

C_(n,k)=((n),(k))=(n!)/((k!)(n-k)!) with $n = \text{population", k="picks}$

And just for fun, the number of possible 5-card poker hands is:

C_(52,5)=((52),(5))=(52!)/((5)!(52-5)!)=(52!)/((5!)(47!))

Let's evaluate it!

(52xx51xxcancelcolor(orange)(50)^10xx49xxcancelcolor(red)48^2xxcancelcolor(brown)(47!))/(cancelcolor(orange)5xxcancelcolor(red)(4xx3xx2)xxcancelcolor(brown)(47!))=52xx51xx10xx49xx2=2,598,960