PERMUTATIONA

Returns the number of ordered permutations, allowing repetition.

Syntax:

PERMUTATIONA(n, k)

where n and k are integers.

PERMUTATIONA returns the number of ordered ways that k objects can be chosen from a set of n objects, where an object can be chosen repeatedly. For example with a set of 3 objects A, B, C, we can choose 2 as follows: AA, AB, AC, BA, BB, BC, CA, CB, CC.

PERMUTATIONA simply calculates $n^{k}$ .

Example:

PERMUTATIONA(3, 2)

returns 9, as in the example above.

Application:

Creating a PIN Code

Imagine you are a security consultant designing a new system for user PIN codes. The PIN must be exactly 4 digits long, and you can use any digit from 0 to 9. You need to determine the total number of possible unique PIN codes that can be created.

In this scenario:

The set of items (n): The digits from 0 to 9. There are 10 possible digits. So, n=10.
The number of choices (r): The PIN code is 4 digits long. You are making 4 choices. So, r=4.
Repetition: You can repeat digits. For example, 1111, 1231, and 5002 are all valid PIN codes.

The Calculation

To find the total number of possible PIN codes, you would use the permutation with repetition formula:

PERMUTATIONA(10, 4)

$n^{r} = 1 0^{4} = 10 \times 10 \times 10 \times 10 = 10, 000$

Table of Choices

You can visualize this process step-by-step using a table to show the number of possibilities for each position in the PIN.

	Digital Position	Number of Choices
	A	B
1	1st Digit	10 (0, 1, 2, ..., 9)
2	2nd Digit	10 (0, 1, 2, ..., 9)
3	3rd Digit	10 (0, 1, 2, ..., 9)
4	4th Digit	10 (0, 1, 2, ..., 9)

Conclusion

Multiplying the number of choices for each position gives you the total number of possible outcomes:

Total PIN codes = (Choices for 1st digit) x (Choices for 2nd digit) x (Choices for 3rd digit) x (Choices for 4th digit) Total PIN codes = 10×10×10×10=10,000

This example demonstrates how the concept of permutations with repetition, often represented by a function like PERMUTATIONA in some software, is used to calculate the total number of possible arrangements when repetition is allowed, such as in creating passwords, license plates, or access codes.

Result for PERMUTATIONA(10, 4):

10,000