What will be the number of permutations of n different things, taken r at a time, where repeatition is allowed?

Question

Prepp Team · Accepted Answer

Permutations with Repetition Explained

The question asks about the number of permutations of 'n' different things, taken 'r' at a time, specifically when repetition is allowed. Understanding this concept is key to solving many combinatorics problems.

Understanding Permutations

A permutation is an arrangement of objects in a specific order. When we talk about permutations, the order of selection matters.

Permutations without Repetition: If you are arranging 'r' items from 'n' distinct items without putting them back (no repetition), the formula is $ P(n, r) = \frac{n!}{(n-r)!} $. For example, arranging 3 letters from ABC without repetition would be ABC, ACB, BAC, BCA, CAB, CBA.
Permutations with Repetition: In this case, after selecting an item, it is put back, allowing it to be selected again for the next position. This means the choices for each position remain constant.

Deriving the Permutation Formula with Repetition

Let's consider how many choices we have for each position when forming an arrangement of 'r' items from 'n' different things, where repetition is allowed.

For the first position, we have 'n' different choices available.
Since repetition is allowed, for the second position, we still have 'n' different choices available.
Similarly, for the third position, we again have 'n' different choices.
This pattern continues for all 'r' positions.

According to the multiplication principle (also known as the fundamental counting principle), if there are $ n_1 $ ways to do the first thing, $ n_2 $ ways to do the second thing, and so on, then there are $ n_1 \times n_2 \times \dots $ ways to do all of them.

In this scenario, we are making 'r' choices, and each choice has 'n' possibilities.

Number of permutations $ = n \times n \times \dots \times n $ (r times)

Therefore, the number of permutations of 'n' different things, taken 'r' at a time, where repetition is allowed, is given by the formula:

$$ n^r $$

Applying the Permutation Concept

Let's use an example to illustrate this. Suppose we have 3 different digits (1, 2, 3) and we want to form a 2-digit number, where repetition of digits is allowed.

Here, $ n = 3 $ (the total number of different digits).
We are taking $ r = 2 $ at a time (forming a 2-digit number).

Using the formula $ n^r $, the number of permutations would be $ 3^2 = 9 $.

The possible 2-digit numbers are:

First Digit	Second Digit	Number Formed
1	1	11
1	2	12
1	3	13
2	1	21
2	2	22
2	3	23
3	1	31
3	2	32
3	3	33

As you can see, there are 9 distinct 2-digit numbers, which matches our calculation of $ 3^2 $.

Analyzing the Given Options

Let's evaluate each of the provided options in the context of permutations with repetition allowed:

Option 1: $ \frac{n}{{r(n-1) }} $
This formula does not correspond to any standard permutation or combination calculation. It is incorrect for the problem described.
Option 2: $ n^{rn} $
This formula suggests 'n' raised to the power of 'r times n'. This value would be significantly larger than expected and does not represent the number of permutations with repetition.
Option 3: $ n^r $
This formula accurately represents the number of permutations of 'n' different things, taken 'r' at a time, where repetition is allowed. As derived above, for each of the 'r' positions, there are 'n' choices.
Option 4: $ (n-1)^r $
This formula implies that the first choice has (n-1) options, which is not true, and it does not account for the availability of all 'n' items for each selection when repetition is permitted. It is incorrect.

Conclusion

Based on the principles of permutations with repetition, the correct formula is $ n^r $. This formula accounts for having 'n' choices available for each of the 'r' selections because items can be repeated.