Number of ways to arrange a word such that all vowels occur together

Difficulty Level : Medium
Last Updated : 06 May, 2021

Given a word containing vowels and consonants. The task is to find that in how many ways to word can be arranged so that the vowels always come together. Given that the length of the word <10.

Examples:

Input: str = "geek"
Output: 6
Ways such that both 'e' comes together are 6 
i.e. geek, gkee, kgee, eekg, eegk, keeg

Input: str = "corporation"
Output: 50400

Approach: Since word contains vowels and consonants together. All vowels are needed to remain together then we will take all vowels as a single letter.

As, in the word ‘geeksforgeeks’, we can treat the vowels “eeoee” as one letter.
Thus, we have gksfrgks (eeoee).
This has 9 (8 + 1) letters of which g, k, s each occurs 2 times and the rest are different.
The number of ways arranging these letters = 9!/(2!)x(2!)x(2!) = 45360 ways
Now, 5 vowels in which ‘e’ occurs 4 times and ‘o’ occurs 1 time, can be arranged in 5! /4! = 5 ways.
Required number of ways = (45360 x 5) = 226800

Below is the implementation of the above approach:

C++

// C++ program to calculate the no. of ways
// to arrange the word having vowels together
#include <bits/stdc++.h>
#define ll long long int
using namespace std;
 
// Factorial of a number
ll fact(int n)
{
    ll f = 1;
    for (int i = 2; i <= n; i++)
        f = f * i;
    return f;
}
 
// calculating ways for arranging consonants
ll waysOfConsonants(int size1, int freq[])
{
    ll ans = fact(size1);
    for (int i = 0; i < 26; i++) {
 
        // Ignore vowels
        if (i == 0 || i == 4 || i == 8 || i == 14 || i == 20)
            continue;
        else
            ans = ans / fact(freq[i]);
    }
 
    return ans;
}
 
// calculating ways for arranging vowels
ll waysOfVowels(int size2, int freq[])
{
    return fact(size2) / (fact(freq[0]) * fact(freq[4]) * fact(freq[8])
                    * fact(freq[14]) * fact(freq[20]));
}
 
// Function to count total no. of ways
ll countWays(string str)
{
 
    int freq[26] = { 0 };
    for (int i = 0; i < str.length(); i++)
        freq[str[i] - 'a']++;
 
    // Count vowels and consonant
    int vowel = 0, consonant = 0;
    for (int i = 0; i < str.length(); i++) {
 
        if (str[i] != 'a' && str[i] != 'e' && str[i] != 'i'
            && str[i] != 'o' && str[i] != 'u')
            consonant++;
        else
            vowel++;
    }
 
    // total no. of ways
    return waysOfConsonants(consonant+1, freq) *
           waysOfVowels(vowel, freq);
}
 
// Driver code
int main()
{
    string str = "geeksforgeeks";
 
    cout << countWays(str) << endl;
 
    return 0;
}

Java

// Java program to calculate the no. of
// ways to arrange the word having
// vowels together
import java.util.*;
  
class GFG{
  
// Factorial of a number
static int fact(int n)
{
    int f = 1;
    for(int i = 2; i <= n; i++)
        f = f * i;
          
    return f;
}
  
// Calculating ways for arranging consonants
static int waysOfConsonants(int size1,
                            int []freq)
{
    int ans = fact(size1);
    for(int i = 0; i < 26; i++)
    {
          
        // Ignore vowels
        if (i == 0 || i == 4 || i == 8 ||
            i == 14 || i == 20)
            continue;
        else
            ans = ans / fact(freq[i]);
    }
    return ans;
}
  
// Calculating ways for arranging vowels
static int waysOfVowels(int size2, int [] freq)
{
    return fact(size2) / (fact(freq[0]) *
          fact(freq[4]) * fact(freq[8]) *
         fact(freq[14]) * fact(freq[20]));
}
  
// Function to count total no. of ways
static int countWays(String str)
{
    int []freq = new int [200];
    for(int i = 0; i < 200; i++)
        freq[i] = 0;
          
    for(int i = 0; i < str.length(); i++)
        freq[str.charAt(i) - 'a']++;
          
    // Count vowels and consonant
    int vowel = 0, consonant = 0;
    for(int i = 0; i < str.length(); i++)
    {
        if (str.charAt(i) != 'a' && str.charAt(i) != 'e' &&
            str.charAt(i) != 'i' && str.charAt(i) != 'o' &&
            str.charAt(i) != 'u')
            consonant++;
        else
            vowel++;
    }
  
    // Total no. of ways
    return waysOfConsonants(consonant + 1, freq) *
           waysOfVowels(vowel, freq);
}
  
// Driver code
public static void main(String []args)
{
    String str = "geeksforgeeks";
  
    System.out.println(countWays(str));
}
}
 
// This code is contributed by rutvik_56

Python3

# Python3 program to calculate
# the no. of ways to arrange
# the word having vowels together
 
# Factorial of a number
def fact(n):
 
    f = 1
    for i in range(2, n + 1):
        f = f * i
    return f
 
# calculating ways for
# arranging consonants
def waysOfConsonants(size1, freq):
 
    ans = fact(size1)
    for i in range(26):
 
        # Ignore vowels
        if (i == 0 or i == 4 or
            i == 8 or i == 14 or
            i == 20):
            continue
        else:
            ans = ans // fact(freq[i])
 
    return ans
 
# calculating ways for
# arranging vowels
def waysOfVowels(size2, freq):
 
    return (fact(size2) // (fact(freq[0]) *
            fact(freq[4]) * fact(freq[8]) *
            fact(freq[14]) * fact(freq[20])))
 
# Function to count total no. of ways
def countWays(str1):
 
    freq = [0] * 26
    for i in range(len(str1)):
        freq[ord(str1[i]) -
             ord('a')] += 1
 
    # Count vowels and consonant
    vowel = 0
    consonant = 0
    for i in range(len(str1)):
 
        if (str1[i] != 'a' and str1[i] != 'e' and
            str1[i] != 'i' and str1[i] != 'o' and
            str1[i] != 'u'):
            consonant += 1
        else:
            vowel += 1
 
    # total no. of ways
    return (waysOfConsonants(consonant + 1, freq) *
            waysOfVowels(vowel, freq))
 
# Driver code
if __name__ == "__main__":
 
    str1 = "geeksforgeeks"
    print(countWays(str1))
 
# This code is contributed by Chitranayal

C#

// C# program to calculate the no. of
// ways to arrange the word having
// vowels together
using System.Collections.Generic;
using System;
 
class GFG{
 
// Factorial of a number
static int fact(int n)
{
    int f = 1;
    for(int i = 2; i <= n; i++)
        f = f * i;
         
    return f;
}
 
// Calculating ways for arranging consonants
static int waysOfConsonants(int size1,
                            int []freq)
{
    int ans = fact(size1);
    for(int i = 0; i < 26; i++)
    {
         
        // Ignore vowels
        if (i == 0 || i == 4 || i == 8 ||
            i == 14 || i == 20)
            continue;
        else
            ans = ans / fact(freq[i]);
    }
    return ans;
}
 
// Calculating ways for arranging vowels
static int waysOfVowels(int size2, int [] freq)
{
    return fact(size2) / (fact(freq[0]) *
          fact(freq[4]) * fact(freq[8]) *
         fact(freq[14]) * fact(freq[20]));
}
 
// Function to count total no. of ways
static int countWays(string str)
{
    int []freq = new int [200];
    for(int i = 0; i < 200; i++)
        freq[i] = 0;
         
    for(int i = 0; i < str.Length; i++)
        freq[str[i] - 'a']++;
         
    // Count vowels and consonant
    int vowel = 0, consonant = 0;
    for(int i = 0; i < str.Length; i++)
    {
        if (str[i] != 'a' && str[i] != 'e' &&
            str[i] != 'i' && str[i] != 'o' &&
            str[i] != 'u')
            consonant++;
        else
            vowel++;
    }
 
    // Total no. of ways
    return waysOfConsonants(consonant + 1, freq) *
           waysOfVowels(vowel, freq);
}
 
// Driver code
public static void Main()
{
    string str = "geeksforgeeks";
 
    Console.WriteLine(countWays(str));
}
}
 
// This code is contributed by Stream_Cipher

Javascript

<script>
// Javascript program to calculate the no. of
// ways to arrange the word having
// vowels together
     
// Factorial of a number
function fact(n)
{
       let f = 1;
    for(let i = 2; i <= n; i++)
        f = f * i;
           
    return f;
}
 
// Calculating ways for arranging consonants
function waysOfConsonants(size1,freq)
{
    let ans = fact(size1);
    for(let i = 0; i < 26; i++)
    {
           
        // Ignore vowels
        if (i == 0 || i == 4 || i == 8 ||
            i == 14 || i == 20)
            continue;
        else
            ans = Math.floor(ans / fact(freq[i]));
    }
    return ans;
}
 
// Calculating ways for arranging vowels
function waysOfVowels(size2,freq)
{
    return Math.floor(fact(size2) / (fact(freq[0]) *
          fact(freq[4]) * fact(freq[8]) *
         fact(freq[14]) * fact(freq[20])));
}
 
// Function to count total no. of ways
function countWays(str)
{
    let freq = new Array(200);
    for(let i = 0; i < 200; i++)
        freq[i] = 0;
           
    for(let i = 0; i < str.length; i++)
        freq[str[i].charCodeAt(0) - 'a'.charCodeAt(0)]++;
           
    // Count vowels and consonant
    let vowel = 0, consonant = 0;
    for(let i = 0; i < str.length; i++)
    {
        if (str[i] != 'a' && str[i] != 'e' &&
            str[i] != 'i' && str[i] != 'o' &&
            str[i] != 'u')
            consonant++;
        else
            vowel++;
    }
   
    // Total no. of ways
    return waysOfConsonants(consonant + 1, freq) *
           waysOfVowels(vowel, freq);
}
 
 
// Driver code
let str = "geeksforgeeks";
document.write(countWays(str));
     
     
    // This code is contributed by avanitrachhadiya2155
</script>

Output:

Further Optimizations: We can pre-compute required factorial values to avoid re-computations.

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