Rr×n

Rr×n is a mathematical concept that represents the number of ways to arrange n distinct objects in a sequence of r objects, where repetition is allowed. This is also known as permutations with repetition. The formula to calculate Rr×n is given by:

Rr×n = n^r

Where:

- n is the number of distinct objects,

- r is the number of objects in each sequence.

This formula is derived from the fact that for each position in the sequence, there are n choices, and this choice is independent of the choices for the other positions.

Rr×n is a fundamental concept in combinatorics, a branch of mathematics that deals with counting, arrangement, and combination of objects. It has numerous applications in various fields, including probability, statistics, and computer science.

For example, if we have 3 distinct objects (n=3) and we want to arrange them in sequences of 2 objects (r=2), the number of possible sequences is Rr×n = 3^2 = 9. The possible sequences are: (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3).

It's important to note that Rr×n is different from the concept of permutations without repetition, where the number of ways to arrange n distinct objects in a sequence of n objects is given by n!, and from combinations, where the order of the objects does not matter.

In summary, Rr×n is a simple yet powerful concept in combinatorics that allows us to calculate the number of ways to arrange n distinct objects in a sequence of r objects with repetition.