# Mathematics | Problems On Permutations | Set 1

Prerequisite – Permutation and Combination

Formula’s Used :

```1. P(n, r) = n! / (n-r)!

2. P(n, n) = n!  ```

Example-1 :
How many 4-letter words, with or without meaning, can be formed out of the letters of the word, ‘GEEKSFORGEEKS’, if repetition of letters is not allowed ?

Explanation :
Total number of letters in the word ‘GEEKSFORGEEKS’ = 13
Therefore, the number of 4-letter words

```= Number of arrangements of 13 letters, taken 4 at a time.
= 13P4 ```

Example-2 :
How many 4-digit numbers are there with distinct digits ?

Explanation :
Total number of arrangements of ten digits ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ) taking 4 at a time

`= 10P4 `

These arrangements also have those numbers which have 0 at thousand’s place.
(For eg- 0789 which is not a 4-digit number.).

If we fix 0 at the thousand’s place, we need to arrange the remaining 9 digits by taking 3 at a time.

Total number of such arrangements

`= 9P3  `

Thus, the total number of 4-digit numbers

`= 10P4  - 9P3`

Example-3 :
How many different words can be formed with the letters of the word “COMPUTER” so that the word begins with “C” ?

Explanation :
Since all the words must begin with C. So, we need to fix the C at the first place.
The remaining 7 letters can be arranged in 7P7 = 7! ways.

Example-4 :
In how many ways can 8 C++ developers and 6 Python Developers be arranged for a group photograph if the Python Developers are to sit on chairs in a row and the C++ developers are to stand in a row behind them ?

Explanation :
6 Python Developers can sit on chairs in a row in 6P6 = 6! ways
8 C++ Developers can stand behind in a row in 8P8 = 8! ways
Thus, the total number of ways

`= 6! x 8! ways `

Example-5 :
Prove that 0! = 1.

Explanation :
Using the formula of Permutation-

```P(n, r) = n! / (n-r)!

P(n, n) = n! / 0!  (Let r = n )

n! = n! / 0!    (Since, P(n, n) = n!)
0! = n! / n!
0! = 1
Thus, Proved ```

Attention reader! Don’t stop learning now. Get hold of all the important CS Theory concepts for SDE interviews with the CS Theory Course at a student-friendly price and become industry ready.

My Personal Notes arrow_drop_up Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Article Tags :

Be the First to upvote.

Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.