Permutations Formula

Permutations Formula
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Have you heard of the famous Arab mathematician, Al-Khalil? He wrote a book titled, "Book of cryptographic messages," and that book contained the first use of Permutations and Combinations.

Permutations are extremely important in Math and they are also essential in various fields of Science such as Physics and Biology.

In this lesson, you will be introduced to the formulas of permutations. You will recall about permutations and learn how to apply the formula to solve problems related to permutations. This lesson includes several simple examples to help you understand the concept clearly. Let's now start this lesson by learning the definition of permutation.

What Is Permutation?

Before discussing the permutation formula, we should understand the concept of “Permutation.”

Defining Permutation: A permutation is an arrangement of a certain number of objects in a specific order.

In fact, permutation is nothing but counting the number of ways in which the given objects can be arranged.

Permutation can be defined as the different ways in which a collection of objects or alphabets can be arranged.

When dealing with permutation, remember that position is important.

Given below are the different formulas of permutation.

1. This formula is used to find the number of possible ways in which anything can be arranged while ensuring that repetition is not allowed.

 P = \begin{align}\frac{n!}{(n-r)!}\end{align}

2. This formula is used to find the number of possible ways in which anything can be arranged while allowing repetition.

 $$P\left ( n,r \right )$$ = $$n^{r}$$

$$n$$ = Total number of elements in a set.

$$r$$ = Number of elements selected.

Did you know that we use the combination formula when the order does not matter? This formula is given as:

 $$C(n,r)$$= \begin{align}\frac{n!}{(n-r)!r!}\end{align}

What Is the General Formula of Permutation?

Formula for Permutation:  $$P$$ = $$\dfrac{n!}{(n-r)!}$$ ; $$0 \leq r \leq n$$

Here “!” represents the product of the integers less than or equal to $$n$$, but it should be greater than or equal to one.

Some common examples are:

$$1!$$ = $$1\times 1$$ = $$1$$

$$2!$$ = $$2\times 1$$ = $$2$$

$$3!$$ = $$3\times 2\times 1$$ = $$6$$

$$4!$$ = $$4\times 3\times 2\times 1$$ = $$24$$

Did you notice that all the numbers are the factors of the result (product)?

 $$P$$ = $$\dfrac{n!}{(n-r)!}$$ ; $$0 \leq r \leq n$$

How to Solve Word Problems Using Permutation?

To solve a word problem using permutation, we will use the general formula based on the information given.

Let us understand factorials using an example.

Represent the factorial for number 4.

Solution: Always remember, factorials are represented with a “!” sign which means 4 will be written as 4!

Now, to find out the value of 4! we will solve it as follows:

$$4!$$ = $$4\times 3\times 2\times 1$$ = 24

Now let us solve a problem using the permutation formula.

Example: Emily is having 4 chairs and she wants to place her three dolls on those chairs. In how many possible ways can Emily place her 3 dolls on 4 chairs?

Solution: Given n = 4 and r = 3

On applying the permutation formula

P = \begin{align}\frac{n!}{(n-r)!}\end{align}

We get,

\begin{align}P &= \dfrac{n!}{(n-r)!}\\ & = \dfrac{4!}{(4-3)!}\\ &= \dfrac{4!}{1!}\\ &= \dfrac{4\times 3\times 2\times 1}{1}\\ &= 24\end{align}

She can place 3 dolls in 24 ways on the 4 chairs.

Important Notes
• If you have five vegetables such as a potato, an onion, a broccoli, a tomato, and an okra that are to be added while cooking a certain dish, but you changed the order of adding these vegetables, it will not make a difference. The dish will taste the same irrespective of the order in which the vegetables were added as long as the vegetables are not replaced by different varieties. Here, "what you add or position" is important and that forms an important aspect of combination.
• If your bag has a lock code, say 345, and you try opening it using 453 or 534, that will not work. You will have to use the numbers 345, and in the same order. Here, "what u choose" is important and that's permutation.

Solved Examples

 Example 1

There are 7 books on a bookshelf.

Can you find out in how many different ways will you be able to rearrange them such that no book can be placed in a spot more than once (repetition is not allowed)?

Solution

Since we have seven books, we will have:

Seven choices for the $$1^{st}$$ book; Six choices for the $$2^{nd}$$ book; Five choices for the $$3^{rd}$$ book; Four choices for the $$4^{th}$$ book; Three choices for the $$5^{th}$$ book; Two choices for the $$6^{th}$$ book; One choice for the $$7^{th}$$ book

Now, multiply the choices: $$7\times 6\times 5\times 4\times 3\times 2\times 1$$ = $$5,040$$

 $$\therefore$$ 5,040 ways are possible
 Example 2

There are 5 pencils in a pencil box.

Help Andrew figure out all the possible ways in which the pencils can be rearranged such that a pencil cannot be placed twice in the same position.

Solution

We have 5 pencils. We know that these pencils cannot be placed twice in the same position.

There will be 5 spot choices for the $$1^{st}$$ pencil; $$4$$ spot choices for the $$2^{nd}$$ pencil; $$3$$ spot choices for the $$3^{rd}$$ pencil; $$2$$ spot choices for the $$4^{th}$$ pencil; $$1$$ spot for the $$5^{th}$$ pencil

Therefore, when we multiply the choices, we get: $$5\times 4\times 3\times 2\times 1$$ = $$120$$

Thus, there are 120 possible arrangements.

 $$\therefore$$ There are 120 possible arrangements.
 Example 3

In how many ways can you rearrange the word “SMILE” so that it may or may not make a valid word?

Note that you cannot place any alphabet more than once at a spot.

Solution

The word “SMILE” has 5 letters.

Hence, the spot choices for these letters will be the same as that in the previous example about 5 pencils.

Therefore, to find out all the possible rearrangements, multiply the choices: $$5\times 4\times 3\times 2\times 1$$ = 120

 $$\therefore$$ 120 ways are possible.
 Example 4

How many 5-digit codes can be created using the digits from 0 to 9 if repetition is allowed?

Solution

Given n = 10 and r = 5

\begin{align}P\left ( n,r \right ) &= n^{r}\\ & =10^{5}\\ &=100,000\end{align}

 $$\therefore$$ 100,000 five-digit codes can be created
 Example 5

Consider a set of 5 alphabets p, q, r, s, t.

In how many ways can 4 alphabets be selected without repetition?

Solution

The set of alphabets are p, q, r, s, t.

4 alphabets are to be selected.

$$\therefore$$  $$p_{r}^{n}$$ = $$p_{4}^{5}$$ = $$\dfrac{n!}{(n-r)!}$$

= $$\dfrac{5!}{(5-4)!}$$

= $$5\times 4\times 3\times 2\times 1$$

= 120

 $$\therefore$$ 120 ways

Challenging Questions
1. Find the sum of all the possible 4-digit numbers formed using the digits 3, 4, 5, and 6 using each digit once?

Interactive Questions

Here are a few activities for you to practice.

Let's Summarize

We hope you enjoyed learning about permutation formula with the examples and practice questions. Now you will be able to easily solve problems on permutation formula with repetition, permutation formula example, and combination formula.

At Cuemath, our team of math experts are dedicated to make learning fun for our favorite readers, the students! Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic. Be it problems, online classes, videos, or any other form of relation, it’s the logical thinking and smart learning approach that we at Cuemath believe in.

1. What is permutation in math?

When it comes to math, permutation can be defined as the process of changing the order or sequence of an already ordered set of objects.

2. What is the difference between permutation and combination?

The most basic difference between permutation and combination is that you have to consider the order in case of permutation, but not when it comes to combination.

The order matters in permutation, but doesn’t matter in the case of a combination.

For example, a mobile phone unlocks with a password pin, which is $$3428$$. If you type $$2438$$, it will not get unlocked because the order of number matters here. That is called a permutation.

If it were a combination, then your phone would unlock by any sequence you make with the numbers, 2, 3, 4 and 8

3. What is “r” in the permutation formula?

The exact formula for permutation is $$\dfrac{n!}{(n-r)!}$$ ; 0 ≤ r ≤ n

Here, while n denotes the total number of items in a set, $$r$$ denotes all the items taken for permutation.

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