Question

What value is a permutation of n things taken r at a time divided by to get the combination of n things taken r at a time?

A. r!
B. (n-r)!
C. r

Answer 1

A. r!

Explanation:

Combinations formula:

`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) = (n!)/(x!(n-x)!)`

Permutations formula:

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

`P_((n,x)) = (n!)/((n-x)!)`

Permutation of n things taken r at a time

`P_((n,r)) = (n!)/((n-r)!)`

Combination of n things taken r at a time:

`C_(n,r) = (n!)/(r!(n-r)!)`

Division:

`(P_((n,r)))/(C_(n,r)) = ((n!)/((n-r)!))/((n!)/(r!(n-r)!)) = (n!)/((n-r)!) * (r!(n-r)!)/(n!) = r!`

So the correct answer is:

A. r!