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

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++

// C++ code for above approach
#include <bits/stdc++.h>
#define ll long long int
using namespace std;

// Function to check if a character is vowel or consonant
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 consonants
    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;
    }

    // return 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);

    // subtract 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;
}

Java

// Java code for above approach
import java.util.*;
import java.lang.*;
class GFG
{
  // Function to check if a character
  // is vowel or consonant
  static boolean 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
  static long fact(long n)
  {
    if (n < 2)
    {
      return 1;
    }
    return n * fact(n - 1);
  }

  // Calculating no of ways for arranging vowels
  static long only_vowels(HashMap<Character, Integer> freq)
  {
    long denom = 1;
    long cnt_vwl = 0;

    // Iterate the map and count the number
    // of vowels and calculate no of ways
    // to arrange vowels
    for (Map.Entry<Character, Integer> itr : freq.entrySet())
    {
      if (isVowel(itr.getKey()))
      {
        denom *= fact(itr.getValue());
        cnt_vwl += itr.getValue();
      }
    }

    return fact(cnt_vwl) / denom;
  }

  // Calculating no of ways to arrange
  // the given word such that all vowels
  // come together
  static long all_vowels_together(HashMap<Character, Integer> freq)
  {

    // Calculate no of ways to arrange vowels
    long vow = only_vowels(freq);

    // To store denominator of fraction
    long denom = 1;

    // Count of consonants
    long cnt_cnst = 0;

    for (Map.Entry<Character, Integer> itr : freq.entrySet())
    {
      if (!isVowel(itr.getKey()))
      {
        denom *= fact(itr.getValue());
        cnt_cnst += itr.getValue();
      }
    }

    // Calculate the number of ways to
    // arrange the word such that
    // all vowels come together
    long ans = fact(cnt_cnst + 1) / denom;

    return (ans * vow);
  }

  // To calculate total number of permutations
  static long total_permutations(HashMap<Character, Integer> freq)
  {

    // To store length of the given word
    long cnt = 0;

    // Denominator of fraction
    long denom = 1;
    for (Map.Entry<Character, Integer> itr : freq.entrySet())
    {
      denom *= fact(itr.getValue());
      cnt += itr.getValue();
    }

    // Return total number of
    // permutations of the given word
    return fact(cnt) / denom;
  }

  // Function to calculate number of permutations
  // such that no vowels come together
  static long no_vowels_together(String word)
  {

    // To store frequency of character
    HashMap<Character, Integer> freq = new HashMap<>();

    // Count frequency of all characters
    for(int i = 0; i < word.length(); i++)
    {
      char ch = Character.toLowerCase(word.charAt(i));
      if (freq.containsKey(ch))
      {
        freq.put(ch, freq.get(ch)+1);
      }
      else
      {
        freq.put(ch, 1);
      }
    }

    // Calculate total number of permutations
    long total = total_permutations(freq);

    // Calculate total number of permutations
    // such that all vowels come together
    long vwl_tgthr = all_vowels_together(freq);

    // Subtract vwl_tgthr from total
    // to get the result
    long res = total - vwl_tgthr;

    // Return the result
    return res;
  }

  // Driver code
  public static void main(String[] args) {
    String word = "allahabad";
    long ans = no_vowels_together(word);
    System.out.println(ans);

    word = "geeksforgeeks";
    ans = no_vowels_together(word);
    System.out.println(ans);

    word = "abcd";
    ans = no_vowels_together(word);
    System.out.println(ans);
  }
}

// This code is contributed by divyesh072019

Python3

# Python3 code for above approach

# Function to check if a character is
# vowel or consonant
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 consonants
    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]

    # return 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 all characters
    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)

    # subtract 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

C#

// C# code for above approach
using System;
using System.Collections.Generic;

class GFG{

// Function to check if a character
// is vowel or consonant
static 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
static long fact(long n)
{
    if (n < 2)
    {
        return 1;
    }
    return n * fact(n - 1);
}

// Calculating no of ways for arranging vowels
static long only_vowels(Dictionary<char, int> freq)
{
    long denom = 1;
    long cnt_vwl = 0;

    // Iterate the map and count the number
    // of vowels and calculate no of ways
    // to arrange vowels
    foreach(KeyValuePair<char, int> itr in freq)
    {
        if (isVowel(itr.Key))
        {
            denom *= fact(itr.Value);
            cnt_vwl += itr.Value;
        }
    }

    return fact(cnt_vwl) / denom;
}

// Calculating no of ways to arrange
// the given word such that all vowels
// come together
static long all_vowels_together(Dictionary<char, int> freq)
{

    // Calculate no of ways to arrange vowels
    long vow = only_vowels(freq);

    // To store denominator of fraction
    long denom = 1;

    // Count of consonants
    long cnt_cnst = 0;

    foreach(KeyValuePair<char, int> itr in freq)
    {
        if (!isVowel(itr.Key))
        {
            denom *= fact(itr.Value);
            cnt_cnst += itr.Value;
        }
    }

    // Calculate the number of ways to
    // arrange the word such that
    // all vowels come together
    long ans = fact(cnt_cnst + 1) / denom;

    return (ans * vow);
}

// To calculate total number of permutations
static long total_permutations(Dictionary<char, int> freq)
{

    // To store length of the given word
    long cnt = 0;

    // Denominator of fraction
    long denom = 1;

    foreach(KeyValuePair<char, int> itr in freq)
    {
        denom *= fact(itr.Value);
        cnt += itr.Value;
    }

    // Return total number of
    // permutations of the given word
    return fact(cnt) / denom;
}

// Function to calculate number of permutations
// such that no vowels come together
static long no_vowels_together(string word)
{

    // To store frequency of character
    Dictionary<char,
               int> freq = new Dictionary<char,
                                          int>();

    // Count frequency of all characters
    for(int i = 0; i < word.Length; i++)
    {
        char ch = Char.ToLower(word[i]);
        if (freq.ContainsKey(ch))
        {
            freq[ch]++;
        }
        else
        {
            freq[ch] = 1;
        }
    }

    // Calculate total number of permutations
    long total = total_permutations(freq);

    // Calculate total number of permutations
    // such that all vowels come together
    long vwl_tgthr = all_vowels_together(freq);

    // Subtract vwl_tgthr from total
    // to get the result
    long res = total - vwl_tgthr;

    // Return the result
    return res;
}

// Driver Code
static void Main()
{
    string word = "allahabad";
    long ans = no_vowels_together(word);
    Console.WriteLine(ans);

    word = "geeksforgeeks";
    ans = no_vowels_together(word);
    Console.WriteLine(ans);

    word = "abcd";
    ans = no_vowels_together(word);
    Console.WriteLine(ans);
}
}

// This code is contributed by divyeshrabadiya07

Javascript

<script>
// Javascript code for above approach

// Function to check if a character
  // is vowel or consonant
function isVowel(ch)
{
    if (ch == 'a' || ch == 'e' ||
        ch == 'i' || ch == 'o' ||
        ch == 'u')
      return true;
    else
      return false;
}

 // Function to calculate factorial of a number
function fact(n)
{
    if (n < 2)
    {
      return 1;
    }
    return n * fact(n - 1);
}

// Calculating no of ways for arranging vowels
function only_vowels(freq)
{
    let denom = 1;
    let cnt_vwl = 0;

    // Iterate the map and count the number
    // of vowels and calculate no of ways
    // to arrange vowels
    for (let [key, value] of freq.entries())
    {
      if (isVowel(key))
      {
        denom *= fact(value);
        cnt_vwl += value;
      }
    }

    return Math.floor(fact(cnt_vwl) / denom);
}

// Calculating no of ways to arrange
  // the given word such that all vowels
  // come together
function all_vowels_together(freq)
{
    // Calculate no of ways to arrange vowels
    let vow = only_vowels(freq);

    // To store denominator of fraction
    let denom = 1;

    // Count of consonants
    let cnt_cnst = 0;

    for (let [key, value] of freq.entries())
    {
      if (!isVowel(key))
      {
        denom *= fact(value);
        cnt_cnst += value;
      }
    }

    // Calculate the number of ways to
    // arrange the word such that
    // all vowels come together
    let ans = Math.floor(fact(cnt_cnst + 1) / denom);

    return (ans * vow);
}

 // To calculate total number of permutations
function total_permutations(freq)
{
    // To store length of the given word
    let cnt = 0;

    // Denominator of fraction
    let denom = 1;
    for (let [key, value] of freq.entries())
    {
      denom *= fact(value);
      cnt += value;
    }

    // Return total number of
    // permutations of the given word
    return Math.floor(fact(cnt) / denom);
}

// Function to calculate number of permutations
  // such that no vowels come together
function no_vowels_together(word)
{
    // To store frequency of character
    let freq = new Map();

    // Count frequency of all characters
    for(let i = 0; i < word.length; i++)
    {
      let ch = word[i].toLowerCase();
      if (freq.has(ch))
      {
        freq.set(ch, freq.get(ch)+1);
      }
      else
      {
        freq.set(ch, 1);
      }
    }

    // Calculate total number of permutations
    let total = total_permutations(freq);

    // Calculate total number of permutations
    // such that all vowels come together
    let vwl_tgthr = all_vowels_together(freq);

    // Subtract vwl_tgthr from total
    // to get the result
    let res = total - vwl_tgthr;

    // Return the result
    return res;
}


// Driver code
let word = "allahabad";
    let ans = no_vowels_together(word);
    document.write(ans+"<br>");

    word = "geeksforgeeks";
    ans = no_vowels_together(word);
    document.write(ans+"<br>");

    word = "abcd";
    ans = no_vowels_together(word);
    document.write(ans+"<br>");

// This code is contributed by unknown2108
</script>

Output:

7200
32205600
0

Time Complexity : O(n) (n = length of string)

Auxiliary Space: O(n), space required for the recursive stack space.