## Permutation "Determining the Number of Permutations"

First, let's make it clear what a permutation is. It is a unique way a number of objects could be ordered or chosen. For example, if we had three letters ABC, we could order them as ABC or BCA. These would be two different permutations. A third permutation would be CAB. What we need to know is how many permutations of these objects are there.

If we have three objects, like three letters, we have 3 possible choices for the first object. This leaves 2 objects and two choices for the second choice, and 1 object for the third choice. So we have 3 times 2 times 1 possible permutation for a result of 6. If we started with 4 objects, there would be 4 times 3 times 2 times 1 or a total of 24 possible permutations. This operation happens often enough for it to have its name and is called the factorial, and the operator is the exclamation point. So $$4! = 4 * 3 * 2 * 1 = 24$$ This is read as **"four factorial"** equals 24.

In general, if we let n be the number of objects, then **n! = n *(n-1)*(n-2)*(n-3)*........ 1**.

Each term in the multiplication is one less than the previous until the value of one is reached. The only exception to this rule is when n=0. In this case, the factorial of 0 is, by definition, equal to 1.

## Theoretical Examples of Permutation

Suppose we only want to select 2 of the 3 objects. In this case, we would have 3 choices for the first object and 2 for the second, for a total of 6 permutations. If we had started with 4 objects and only wanted to select 2, the number of permutations would have been 4 for the first choice and 3 for the second choice for a total of 12 permutations.

In the above two examples, to find the number of permutations of n objects taken 2 at a time, we multiplied the first 2 numbers of the factorial. That is, for 3 we used 3 and 2, and for 4 we used 4 and 3. Let's focus on the case of 4 objects taken 2 at a time. In this example, we used the first two numbers, 4 and 3 of 4! The remaining numbers of 4! Are 2 and 1 or 2! In general, for n objects n! Divided by (n-k)! Where k is the number of objects, we take from the total of n objects. So for this example 4! Divided by $$(4-2)!$$ Which is $$\frac{4*3*2*1}{2*1}$$ and it equals 12.

## Permutation Formula

As we discussed earlier, if an event occurs in **"m"** different ways, following which another event can occur in **"n"** different ways, then the total number of events which occurs is **'m * n'**. This is called the fundamental principle of counting and permutation is based on this principle.

Let's consider an example of the fundamental principle of counting to make it easy for you to understand the formula of permutation next.

**Ex:** Suppose you have three shirts and two pants, in how many combinations are possible. You can select any shirt from three shirts and pants from three pants.

**Answer:** The total possible combinations will be 6 (3x2=6) in this case.

A permutation is an arrangement in a definite order of the number of objects taken some or all at a time but within a defined order. So, now based on the definition, let's derive the formula of permutation.

## Derivation of Permutation's Formula

Number of permutations **"n"** different objects taken **"r"** at a time is:

$$\text{npr}$$

$$\text{Where}\;0\;<\;r\;<=\;n$$

$$npr = \frac{n!}{(n-r)!}$$

## Mathematical Examples of Permutation

Now let's consider some numerical examples of permutation to understand its formula.

**Example 1: **

How many different signals can be made by 3 flags out of 4 flags of different colors?

According to the formula of permutation, here **n=4** and **r=3** as we need to make a combination of 3 flags out of 4 flags. Therefore

$$4p3 = \frac{4!}{(4-3)!} = \frac{24}{1} = 24$$

So there are 24 signals which can be made by 3 flags from 4 flags of different colours.

So the number of permutations of n objects taken k at a time is $$\frac{n!}{(n-k)!}$$That is n factorial divided by n minus k factorial; remember k is the number of objects we're taking from n objects.

Let's do one more example and start with 5 objects and take 3 at a time.

**Example 2: **

$$5p3 = \frac{5!}{(5-3)!} = 60$$

For the numerator that would be 5! And for the denominator, it would be 5 minus 3 factorial or 2 factorial. So the expression is 5! Divided by 2! This reduces to 5*4*3 and equals 60 permutations of 5 objects taken 3 at a time.

## Permutation Calculator

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## How to Calculate Permutations using an Online Permutation Calculator?

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- Select the number of permutations you want to calculate.
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