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

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

• A. (n!)/(n-r)!
• B. n!
• C. (n!)/((n-r)!r!)
• D. n(n-1)(n-2)...(2)(1)

Answer: (A)

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

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