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# Permutation

Updated on January 6, 2024
Article byPrakhar Gajendrakar
Edited byCollins Enosh
Reviewed byDheeraj Vaidya, CFA, FRM

## Permutation Meaning

A permutation is the total number of ways a sample population can be arranged. It is a mathematical calculation used for data sets that follow a particular order. Permutation differs from combinations; they are two different mathematical techniques.

For eg:
Source: Permutation (wallstreetmojo.com)

It is further classified into four types—repetitive, non-repetitive, circular, or multisets. This mathematical tool is extensively used in research, calculation, scientific analysis, and programming—especially in python.

### Key Takeaways

• A permutation is the number of ways a particular data set or sample can be arranged or rearranged. It follows a particular order or sequence.
• It differs from a combination; multiple scenarios can be determined from a single combination.
• Typically, this method is used for large data sets; the researcher need not record each possibility separately.

### Permutation Explained

A permutation is a mathematical formula; it is used to determine the number of ways a sample population can be arranged. Typically, this formula is used for large data sets; the researcher need not record each possibility separately.

To understand the concept, let us assume that there are three pens (P1, P2, P3) and two markers (M1, M2). Every item is distinct; no two items are the same. What are the different ways of picking just one item of the lot? We can pick the first pen (P1), the second pen, maybe the third pen, or maybe even a marker. There are five ways in which we can pick any one item ((P1, P2, P3, M1, M2).

How many arrangements can we pick one pen and marker instead of any item? We can pick pen 1 and marker 1 (P1M1), or we can pick P1M2. Again, it is obvious that there are six different scenarios or six different permutations—P1M1, P1M2, P2M1, P2M2, P3M1, and P3M2.

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### Types

There are four types of permutations –

• Repetitive – In this type, different choices are made each time different objects are selected (from the same data set). Therefore, the same data is used multiple times.
• Non-repetitive – every time a permutation is determined, one item from the data set is eliminated, and the selection continues with available data.
• Multiset – It uses nondistinct samples from a data set. Multisets are less common.
• Circular permutation – Data samples are arranged around a circle or cyclical order in this method.

### Formula

The permutation formula is as follows:

nPr = (n!) / (n-r)!

Here,

• nPr denotes permutation.
• n denotes the number of objects.
• r denotes the number of objects selected.
• ! denotes the factorial.

### Example

Now, let us look at a permutation example to understand it better.

Let us assume that there are nine students in a class. Two of these students are Lois and Clark. Lois and Clark do not get along with each other. Now, let us determine the number of ways the teacher can arrange the students without resulting in Lois and Clark sitting together.

The total number of possible arrangements is 9P9 = 9!

Here, “!” denotes factorial. That is, the factorial of a non-negative n! is the product of all positive integers less than or equal to n.

Now, let us label the number of arrangements in which Lois and Clark are together as LC (treating Lois and Clark as a single unit). The teacher is left with a total of 8 entities to permute: the seven students and LC. These 8 entities can be permuted in 8P8 = 8! ways.

But for each of the 8! permutations, Lois and Clark can be permuted among themselves, as LC and CL. Thus, the total number of permutations in which Lois and Clark are together is “2 × 8!.”

Therefore, the number of permutations in which Lois and Clark are not together is as follows:

• nPr = 9! – (2 × 8!)
• nPr = 362880 – (2 x 40320)
• nPr = 282240

### Importance

The importance of permutation is as follows:

• It is used in statistical analysis and hypothesis testing.
• It offers quick and effective results. It saves time and effort when data sets are large.
• Programming languages use it for coding and calculation.
• The permutation matrix is used in advanced scientific research.

### Difference Between Permutation And Combination

• Permutation refers to all the possible ways a data set can be arranged in a sequence. In contrast, a combination can opt from a data set.
• For the former, the order is given relevance. In the latter, relevance must be provided with the data set.
• The former refers to the process of arrangement. The latter is used for selection.
• The former focuses on ordered elements; the latter does not.
• Multiple permutations are possible from a single combination. On the contrary, only one combination comes from a permutation.

### Permutation vs Factorial vs Bootstrap

• Permutations focus on the sequence, but the factorial involves all possible results in an event. In contrast, bootstrap quantifies data and distribution.
• Permutation and factorial are used for hypothesis testing, whereas bootstrap estimates data intervals.
• A permutation is based on assumptions that may or may not be relevant, but bootstrap determines the variability of the process; it is an in-depth data interval study.

1. Which situation illustrates permutation?

It is used in real-world scenarios, mathematics, and scientific analysis—whenever there is a requirement to set or arrange things in order. This method prioritizes arrangement or placement in a particular order. For example, prize winners can be seated in the order of their ranks.

2. What is the easiest way to calculate permutations?

Every event has a set number of possibilities; we multiply it by the number of events. For example, with four-digit values, each digit can range from 0 to 9, providing ten possibilities for each digit.

3. When are permutations and combinations used?

The former is used when a particular order is required. When a particular order is not required, researchers go with the latter—combination.

This article has been a guide to Permutation and its meaning. Here, we explain its types, formula, example, importance, and comparison with factorial and bootstrap. You can learn more about it from the following articles –