How many possible ways you can make your 4 digit PIN if there is no number repeated?

Video Transcript

This problem, it is said that you are creating a 4-digit pin code. We need to determine how many choices there are. Now. In the first case it is said that we have no restrictions. So we want a 4-digit pin code. So there are a total of four digits and there are no restrictions. So that means we have 10 options to choose from for the first digit because we have a total of 10 digits, 012345678 and nine. And there are no restrictions. So there are 10 options for the second digit, 10 options for the third digit and 10 options for the four digit as well. Using the multiplication rule of counting, if we multiply these numbers, we will have the number of ways of selecting these four visits. And so we'll have the total number of choices. So we end up with 10,000. So there are a total of 10,000 choices in this case. The next problem, Once again we have four digits. However, this time it is said that no digit is repeated. So that means for the first digit we have 10 options to choose from. We can choose any digit we want from 0-9. And for the second digit we can choose any digit except for the one which we chose over here because no digit can be repeated. So that's 10 -1. A total of nine options to choose from. For the third digit. We can choose any digit except for the two digits over here. So we have 10 -2. A total of eight options. Similarly for the four digit we can choose any digit except for the three digits we chose In these three places. So that's 10 -3. A total of seven options. So using the multiplication rule of counting. If we multiply these numbers will have the number of ways of selecting the four digits so that no digit is repeated and that will give us the number of choices. If we calculate business is equal to 5040. So we have 5040 different choices. The next part once again we have four digits And here it is said that no digit is repeated and two and 5 must be present. So know that two and five must be present. So that's two of the four digits two and five because no digit can be repeated. The remaining two digits will have to be any of the digits except for two and five. So there are a total of 10 digits from 0 to 9 and we can't use two and five anymore. So that means there are eight digits to choose from and out of them. We need to select the remaining two digits. We can do that in eight c. Two ways here. Well you see which represents combination and we use combination and not permutation in this case because we are not considering the order just yet. We're just selecting any two digits out of the remaining eight. So now we have all of our required digits, we have to we have a five and we have two more digits out of the remaining eight. Now we need to arrange these four selected digits amongst themselves and that can be done in four factorial ways because we can arrange n objects amongst themselves in n factorial ways. So next we use the multiplication rule of counting and multiply these two and this will give us our required number of choices. So eight C two is eight factorial by two factorial times. The factorial of eight minus two, it's a six And here we have four factorial. So if we calculate this we will end up with 672. So we have a total of 672 different choices.

Introduction

When we hear the word "combination" in our daily life, we immediately think about the collection of things in the form of a set or a group. For instance, if anyone says that my bowl has a combination of apples, carrots, and bananas, then we immediately think that the bowl has three items. We are not concerned with the order in which these three things were put in the bowl.

In mathematics, the combination means the number of ways in which different objects are combined to form a set. The order of elements is not important in a combination. We always study combination with permutation in mathematics because there are many similarities between these two terms. The primary difference between the combination and permutation is that the order matters in permutation while it does not matter in combination. In other words, we can say that the permutation is an ordered combination.

You have already read an example of a simple combination above when three things are put in a bowl. Now, let us consider another scenario.

Harry wants to make a pin code by choosing 4 digits from the set of first five whole numbers (0,1,2,3,4). Suppose his chosen pin code is 4013. Can he rearrange the digits as 3014 or 0143 etc.?

Of course not, the order of the digits is important. If the order of the digits is changed, then the pin code will not work. It means that the selection of code from the first five whole numbers is an example of the permutation.

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Let's go

Types of Permutations

There are two types of permutations:

• Permutations with repetition
• Permutations without repetition

In this article, we will specifically discuss permutation with repetition.

We know that in the permutations, the order of elements is important. Permutations with repetition mean we can select one item twice. The formula for computing the permutations with repetitions is given below:

Here:

n = total number of elements in a set

k = number of elements selected from the set

Consider the following example:

From the set of first 10 natural numbers, you are asked to make a four-digit number. How many different permutations are possible?

Here, first, we need to determine whether we can choose a digit twice or not. We can have four-digit numbers such as 1000, 1002, 3032, and 4044. In all these numbers, one digit is repeated twice or thrice. Therefore, it means that it is an example of permutations with repetition.

The total number of elements in a set is 10 and the number of digits we want to select from this set is 4. Therefore, we will get permutations by substituting the values in the following formula:

Hence, 10000 permutations are possible if we want to make a four-digit number from the set of the first 10 natural numbers.

Sometimes we are given a problem in which the identical items of type 1 are repeated  "p" number of times, type 2 are repeated "q" number of times, type 3 are repeated "r" number of times, and so on. The question arises what shall we do in this case? Well, the answer is simple. There is a separate formula to compute permutations in such problems.

Since the items are repeated, therefore such scenarios are also examples of permutations with repetition. The formula that should be used while computing the permutations in such cases is given below:

Let us solve the following example through the above formula to make the whole concept clearer.

How many eight-digit numbers can be formed with the numbers 2, 2, 2, 3, 3, 3, 4, 4?

Here, n = 8, p = 3, q = 3, and r = 2.

In this example, the order of elements matter, and digits are repeated.  We will substitute the above values in the formula below:

Hence, 560 permutations are possible.

Let us solve some more examples below:

Example 1

In how many ways can the alphabets of the word EXCELLENT  be arranged?

Solution

Total number of elements in the word = n = 9

E is repeated three times, hence p = 3

L is repeated 2 times, hence q = 2

Substitute these values in the formula below to get the number of ways in which the letters of this word can be arranged:

Hence, the letters in the word EXCELLENT can be arranged in 30240 ways.

Example 2

A ship must raise eight flags at one time (two red, two blue, and four green). How many different flag combinations can be raised at a time?

Solution

Here:

The total number of flags = n = 8

Number of red flags = p = 2

Number of blue flags = q = 2

Number of green flags = r = 4

This is an example of permutation with repetition because the elements of the set are repeated and their order is important.

Put the above values in the formula below to get the number of permutations:

Hence, flags can be raised in 420 ways.

Example 3

John owns six-colored pair of shoes (two red, two blue and two black). He wants to put all these pairs of shoes on the shoe rack. How many different arrangements of shoes are possible?

Solution

Here:

The total number of pair of shoes = n = 6

Number of red shoes = p = 2

Number of blue shoes = q = 2

Number of back shoes = r = 2

This is an example of permutation with repetition because the elements are repeated and their order is important.

Put the above values in the formula below to get the number of permutations:

Hence, shoes can be arranged on the shoe rack in 90 ways.

Example 4

In how many ways the alphabets of the word ELECTRIC can be arranged?

Solution

Total number of elements in the word = n = 8

E is repeated three times, hence p = 2

C is repeated 2 times, hence q = 2

Substitute these values in the formula below to get the number of ways in which the letters of this word can be arranged:

Hence, the letters in the word ELECTRIC can be arranged in 10080 ways.

Example 5

A person has to choose three-digits from the set of following seven numbers to make a three-digit number.

{1, 2, 3, 4, 5, 6, 7}

How many different arrangements of the digits are possible?

Solution

A three-digit number can have 2 or three identical numbers. Similarly, in a number, the order of digits is important.

It is given that the person can select 3 digits from the set of 7 numbers. Hence, n = 7 and k = 3. Substitute these values in the formula below to get the number of possible arrangements.

Hence, 343 different arrangements are possible.

How many combinations can you make with 4 numbers without repeating?

How many different possibilities are there for a 4

How many 4