Which formula represents the calculation of permutations of n objects taken r at a time?

The formula for calculating the permutations of ( n ) objects taken ( r ) at a time is represented as ( \frac{n!}{(n-r)!} ). This formula arises from the need to count the arrangements of ( r ) items chosen from a total of ( n ) distinct items, where the order of selection matters.

To understand why this formula works, consider that when selecting the first object, you have ( n ) options available. Once that first object is selected, you have ( n-1 ) options left for the second object, ( n-2 ) for the third, and so forth, until you have selected ( r ) objects. Thus, you multiply the choices:

[

n \times (n-1) \times (n-2) \times \ldots \times (n-r+1).

]

This product can be expressed in terms of factorials. Specifically, the total number of arrangements of ( n ) objects is ( n! ), which accounts for all possible orderings of these ( n ) items. However, after selecting ( r ) items, the order of the remaining ( n-r ) items does not