Are you always confused between permutation and combination? You don’t need to worry anymore as we provide all the details in which you can understand the topic better. Let’s see what permutation and combination is and try to understand their definition first.

Permutation and combination are the ways in which a group of objects are represented by selecting them in a set and forming subsets. These objects are usually done without replacement. The selection of subsets is called permutation when the order of selection is a factor and combination is

when order is not a factor. These are both used as important parts of counting and they are very important in Mathematics as it helps the students enhance their knowledge. In the 17 th century, the French mathematicians Blaise Pascal and Pierre de Fermat gave impetus to the development of combinatorics and probability theory. Both sound complicated, isn’t it?

Permutation is a selection process in which the order matters. It relates to the act of arranging all the members of a set into some sequence. If the set is already ordered, then the rearranging of its elements is called the process of permuting. Every detail matters in permutation. They occur in more or less prominent ways, in almost every area of life. It is denoted by n P r

## Permutation with Repetition

Selecting r of something that has n different types, the permutation will be:

n x n x …r times

so, the general formula is n r

were,

n = the number of elements to choose from

r= the number of times

for example, there are 10 numbers to choose from (0,1,2,3,4,5,6,7,8,9) and we choose 4 of them

so, we write 10 x 10 x … (4 times) = 10 4 = 10000 permutations

## Permutation without Repetition

Example: In how many ways may 3 books be placed next to each other on a shelf?

Since there are 3 books, the first place may be filled in 3 ways. There are then 2 books left, and the second place may be filled in 2 ways. There is only 1 book left to fill the last place.

Since the arrangement of books on the shelf is important, it is a permutations problem.

P 3 = 3! = 3 x 2 x 1 = 6

Thus, the books can be arranged in 6 ways.

During permutation without repetition, our choices get reduced each time.

The formula that is used here is

= n! / (n-r)!

Combination is a way of selecting items from a collection, such that the order of selection does not matter. It is the way of selecting items from a bulk collection and the order in which it is arranged is least looked at. We can count the number of combinations in smaller cases. Combination refers to the combination of n things taken k at a time without repetitions. To refer to combinations in which repetition is allowed, the terms k-selection or k-combination with repetition are often used.

Combination is the choice of r things from a set of n things without replacement and where order doesn’t matter. It is denoted by n C r

= n! /r! (n-r)

### Example 1

A person is going to a candy shop where there are 8 types of flavors, if this person is only going to buy 3, define every combination possible

So, it is a sum of with repetition and the formula

nCr= (n+r-1)! / [r! *(n-1)!]

8C3 = (8+3-1)/ [3! *(8-1)!]

8C3 = (10)! / [3! *7!] 8C3 = (10*9*8*7!)/ (3! *7!)

8C3 = 720/6

8C3 = 120 (here 7! Gets cancelled in the numerator and denominator)

### Example 2

2 girls will go to a party, if between the two, they have 4 pairs of fancy shoes, define the

combination of shoes these two girls can wear

So, this sum without repetition,

Hence, 4C2 = n! /[(n-r)! *r!]

4C2 = 4! / [(4-2)! *2!]

4C2= (4*3*2*1)/ (2*1 * 2*1)

4C2= 24/4

4C2= 6

## Difference between permutation and combination

### Permutations

- Arrangement of people, numbers, alphabets, letters, etc.
- Picking two favorite colors, in order, from a color book.
- Picking first, second and third prize winners.

### Combinations

- Selections of the food, clothes, subjects, team, etc.
- Picking two colors from a color book
- Picking three winners only

### Easy to Remember

- When the order doesn’t matter, it is a combination.
- When the order does matter, it is a permutation.

### For example

- My fruit salad is a combination of mangoes, grapes, and apples. We don’t care what order the fruits are in, they could also be “grapes, mangoes and apples” or “apples, mangoes and grapes”, it is the same fruit salad. This is a combination.
- The password to the account is 874. Now this is important as we can only open the account when we type 874 in order. If we type 784, the account will never open. So, this is a permutation.

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