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This tool generates various questions for finding the number of arrangements using the letters of a given word under different conditions. It also provides mathematical solution for all such questions generated.

For example, enter the word 'UNDERSTNADING' and click on 'GENERATE'. It generates questions based on the arrangements of letters of the word 'UNDERSTNADING' with solutions.

'Permutation and Combination' is an important topic for many competitive exams. There is always a category of questions asking to find the number of arrangements possible (or the number of words with or without meaning that can be formed) using letters of a word under different conditions as follows.

- without any restrictions
- all the vowels (or consonants) come together
- all the vowels (or consonants) never come together
- no two vowels (or consonants) come together
- position of a particular letter remains unchanged
- a particular letter always occur at a particular place
- must start and end with specified letters
- all the occurrences of a particular letter must come together
- no two occurrences of a particular letter can be together
- vowels (or consonants) must occupy only even (or odd) positions
- relative position of the vowels and consonants remains unaltered
- with exactly two (or three, four etc) adjacent vowels (or consonants)
- always two (or three, four etc) letters between two occurrences of a particular letter
- at least two vowels (or consonants) come together
- one particular letter must come before another particular letter
- two particular letters are not next to each other
- two particular letters must always come together
- start with a vowel (or consonant)
- start with a vowel (or consonant) and end with a vowel (or consonant)
- always two (or three, four etc) letters between two particular letters
- one letter must come before another letter and this second letter must come before a third letter
- taking two (or three, four etc) letters at a time
- no two occurrences of a particular letter can come together and no two occurrences of another particular letter can come together.
- no two adjacent letters are alike

This tool helps to practise such questions. For example, for an input word 'LEADING', one of the questions and its solution generated by this tool is given below

__In how many different ways can we arrange four letters of the word LEADING ?__

The word 'LEADING' has $7$ different letters.

$4$ letters can be selected from these $7$ letters in $7\text{C}4$ ways. These $4$ letters can be arranged among themselves in $4!$ ways.

Therefore, required number of ways

$=7\text{C}4×4!=840$

$4$ letters can be selected from these $7$ letters in $7\text{C}4$ ways. These $4$ letters can be arranged among themselves in $4!$ ways.

Therefore, required number of ways

$=7\text{C}4×4!=840$

To programmatically generates all these $840$ arrangements, Permutations Generator can be used.

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