How Do Permutations Differ from Combinations in Mathematics?

[kasse]

What's the difference between these two:

1) The number of permutations of n distinct objects taken r at a time is $$\frac{n!}{(n-r)!}$$

and

2) The number of combinations of n distinct objects taken r at a time is $$\frac{n!}{r!(n-r)!}$$

?

 

[statdad]

Both ideas deal with counting the number of ways to make selections. For the formulas you have,

Permutations
* You have a collection of $$n$$ distinct items
* You select $$r$$ of them without replacement
* You are concerned with the order of selection

Combinations
* You have a collection of $$n$$ distinct items
* You select $$r$$ of them without replacement
* You are not concerned with the order of selection

Suppose your set is [tex]\{a, b, c, d\}<br /> <br /> The number of permutations of 2 things taken from this group is $$12$$. They are (order is first selected, second selected)<br /> a, b<br /> a, c<br /> a, d<br /> b, a<br /> b, c<br /> b, d<br /> c, a<br /> c, b<br /> c, d<br /> d, a<br /> d, b<br /> d, c<br /> <br /> The number of combinations of two things taken from this group is $$6$$. They are<br /> a,b<br /> a,c<br /> a,d<br /> b,c<br /> b,d<br /> c,d<br /> <br /> Think this way: combinations count subsets - order is not important.[/tex]