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\}

The number of permutations of 2 things taken from this group is $$12$$. They are (order is first selected, second selected)
a, b
a, c
a, d
b, a
b, c
b, d
c, a
c, b
c, d
d, a
d, b
d, c

The number of combinations of two things taken from this group is $$6$$. They are
a,b
a,c
a,d
b,c
b,d
c,d

Think this way: combinations count subsets - order is not important.