# Number of ways to arrange a word such that no vowels occur together

Given an English word of length at most 20 characters. Calculate the number of ways to arrange the word such that no vowels occur together.

Note : If the total number of vowels in the given word is one then the result should be 0.

Examples:

```Input : allahabad
Output : 7200

Input : geeksforgeeks
Output : 32205600

Input : abcd
Output : 0
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Since the word contains vowels and consonants. Calculate the total number of ways to arrange the given word and subtract the number of ways having all vowels together. To calculate the total number of ways we’ll use the following formula-

```No of ways = (n!) / (r1! * r2! * ... * rk!)
```

Where n is the number of different characters in the word and r1, r2 … rk, are the frequency of same type character.

In order to calculate the number of ways such that all vowels occur together, we consider the group of all vowels as a single character and using the above formula we’ll calculate the total number of ways having all vowel together. Now subtract it from the total number of ways to get the result.

Below is the implementation of the above approach:

## C++

 `#include ` `#define ll long long int ` `using` `namespace` `std; ` ` `  `// Function to check if a character is vowel or consonent ` `bool` `isVowel(``char` `ch) ` `{ ` `    ``if` `(ch == ``'a'` `|| ch == ``'e'` `|| ch == ``'i'` `|| ch == ``'o'` `|| ch == ``'u'``) ` `        ``return` `true``; ` `    ``else` `        ``return` `false``; ` `} ` ` `  `// Function to calculate factorial of a number ` `ll fact(ll n) ` `{ ` `    ``if` `(n < 2) ` `        ``return` `1; ` `    ``return` `n * fact(n - 1); ` `} ` ` `  `// Calculating no of ways for arranging vowels ` `ll only_vowels(map<``char``, ``int``>& freq) ` `{ ` `    ``ll denom = 1; ` `    ``ll cnt_vwl = 0; ` ` `  `    ``// Iterate the map and count the number of vowels and calculate ` `    ``// no of ways to arrange vowels ` `    ``for` `(``auto` `itr = freq.begin(); itr != freq.end(); itr++) { ` `        ``if` `(isVowel(itr->first)) { ` `            ``denom *= fact(itr->second); ` `            ``cnt_vwl += itr->second; ` `        ``} ` `    ``} ` ` `  `    ``return` `fact(cnt_vwl) / denom; ` `} ` ` `  `// calculating no of ways to arrange the given word such that all vowels ` `// come together ` `ll all_vowels_together(map<``char``, ``int``>& freq) ` `{ ` `    ``// calculate no of ways to arrange vowels ` `    ``ll vow = only_vowels(freq); ` ` `  `    ``// to store denominator of fraction ` `    ``ll denom = 1; ` ` `  `    ``// count of consonents ` `    ``ll cnt_cnst = 0; ` ` `  `    ``for` `(``auto` `itr = freq.begin(); itr != freq.end(); itr++) { ` `        ``if` `(!isVowel(itr->first)) { ` `            ``denom *= fact(itr->second); ` `            ``cnt_cnst += itr->second; ` `        ``} ` `    ``} ` ` `  `    ``// calculate the number of ways to arrange the word such that ` `    ``// all vowels come together ` `    ``ll ans = fact(cnt_cnst + 1) / denom; ` ` `  `    ``return` `(ans * vow); ` `} ` ` `  `// To calculate total number of permutations ` `ll total_permutations(map<``char``, ``int``>& freq) ` `{ ` `    ``// To store length of the given word ` `    ``ll cnt = 0; ` ` `  `    ``// denominator of fraction ` `    ``ll denom = 1; ` ` `  `    ``for` `(``auto` `itr = freq.begin(); itr != freq.end(); itr++) { ` `        ``denom *= fact(itr->second); ` `        ``cnt += itr->second; ` `    ``} ` ` `  `    ``// retur total number of permutations of the given word ` `    ``return` `fact(cnt) / denom; ` `} ` ` `  `// Function to calculate number of permutations such that ` `// no vowels come together ` `ll no_vowels_together(string& word) ` `{ ` `    ``// to store frequency of character ` `    ``map<``char``, ``int``> freq; ` ` `  `    ``// count frequency of all characters ` `    ``for` `(``int` `i = 0; i < word.size(); i++) { ` `        ``char` `ch = ``tolower``(word[i]); ` `        ``freq[ch]++; ` `    ``} ` ` `  `    ``// calculate total number of permutations ` `    ``ll total = total_permutations(freq); ` ` `  `    ``// calculate total number of permutations such that ` `    ``// all vowels come together ` `    ``ll vwl_tgthr = all_vowels_together(freq); ` ` `  `    ``// substrat vwl_tgthr from total to get the result ` `    ``ll res = total - vwl_tgthr; ` ` `  `    ``// return the result ` `    ``return` `res; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` ` `  `    ``string word = ``"allahabad"``; ` `    ``ll ans = no_vowels_together(word); ` `    ``cout << ans << endl; ` ` `  `    ``word = ``"geeksforgeeks"``; ` `    ``ans = no_vowels_together(word); ` `    ``cout << ans << endl; ` ` `  `    ``word = ``"abcd"``; ` `    ``ans = no_vowels_together(word); ` `    ``cout << ans << endl; ` `    ``return` `0; ` `} `

## Python3

 `# Python3 code for above approach ` ` `  `# Function to check if a character is ` `# vowel or consonent ` `def` `isVowel(ch): ` `    ``if` `(ch ``=``=` `'a'` `or` `ch ``=``=` `'e'` `or`  `        ``ch ``=``=` `'i'` `or` `ch ``=``=` `'o'` `or` `ch ``=``=` `'u'``) : ` `        ``return` `True` `    ``else``: ` `        ``return` `False` ` `  `# Function to calculate factorial of a number ` `def` `fact(n): ` `    ``if` `(n < ``2``): ` `        ``return` `1` `    ``return` `n ``*` `fact(n ``-` `1``) ` ` `  `# Calculating no of ways for arranging vowels ` `def` `only_vowels(freq): ` ` `  `    ``denom ``=` `1` `    ``cnt_vwl ``=` `0` ` `  `    ``# Iterate the map and count the number of  ` `    ``# vowels and calculate no of ways to arrange vowels ` `    ``for` `itr ``in` `freq: ` `        ``if` `(isVowel(itr)): ` `            ``denom ``*``=` `fact(freq[itr]) ` `            ``cnt_vwl ``+``=` `freq[itr] ` ` `  `    ``return` `fact(cnt_vwl) ``/``/` `denom ` ` `  `# calculating no of ways to arrange the given word  ` `# such that vowels come together ` `def` `all_vowels_together(freq): ` `     `  `    ``# calculate no of ways to arrange vowels ` `    ``vow ``=` `only_vowels(freq) ` ` `  `    ``# to store denominator of fraction ` `    ``denom ``=` `1` ` `  `    ``# count of consonents ` `    ``cnt_cnst ``=` `0` ` `  `    ``for` `itr ``in` `freq: ` `        ``if` `(isVowel(itr) ``=``=` `False``): ` `            ``denom ``*``=` `fact(freq[itr]) ` `            ``cnt_cnst ``+``=` `freq[itr] ` ` `  `    ``# calculate the number of ways to arrange  ` `    ``# the word such that vowels come together ` `    ``ans ``=` `fact(cnt_cnst ``+` `1``) ``/``/` `denom ` ` `  `    ``return` `(ans ``*` `vow) ` ` `  `# To calculate total number of permutations ` `def` `total_permutations(freq): ` `     `  `    ``# To store length of the given word ` `    ``cnt ``=` `0` ` `  `    ``# denominator of fraction ` `    ``denom ``=` `1` ` `  `    ``for` `itr ``in` `freq: ` `        ``denom ``*``=` `fact(freq[itr]) ` `        ``cnt ``+``=` `freq[itr] ` ` `  `    ``# retur total number of permutations  ` `    ``# of the given word ` `    ``return` `fact(cnt) ``/``/` `denom ` ` `  `# Function to calculate number of permutations  ` `# such that no vowels come together ` `def` `no_vowels_together(word): ` `     `  `    ``# to store frequency of character ` `    ``freq ``=` `dict``() ` ` `  `    ``# count frequency of acharacters ` `    ``for` `i ``in` `word: ` `        ``ch ``=` `i.lower() ` `        ``freq[ch] ``=` `freq.get(ch, ``0``) ``+` `1` ` `  `    ``# calculate total number of permutations ` `    ``total ``=` `total_permutations(freq) ` ` `  `    ``# calculate total number of permutations  ` `    ``# such that vowels come together ` `    ``vwl_tgthr ``=` `all_vowels_together(freq) ` ` `  `    ``# substrat vwl_tgthr from total ` `    ``# to get the result ` `    ``res ``=` `total ``-` `vwl_tgthr ` ` `  `    ``# return the result ` `    ``return` `res ` ` `  `# Driver code ` `word ``=` `"allahabad"` `ans ``=` `no_vowels_together(word) ` `print``(ans) ` ` `  `word ``=` `"geeksforgeeks"` `ans ``=` `no_vowels_together(word) ` `print``(ans) ` ` `  `word ``=` `"abcd"` `ans ``=` `no_vowels_together(word) ` `print``(ans) ` ` `  `# This code is contributed by Mohit Kumar `

Output:

```7200
32205600
0
```

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