Arrangement of the characters of a word such that all vowels are at odd places

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Given a string ‘S’ containing vowels and consonants of lowercase English alphabets. The task is to find the number of ways in which the characters of the word can be arranged such that the vowels occupy only the odd positions.

Examples:

Input: zambiatek
Output: 36
$3_P_2 \times 3_P_3 = 36$

Input: publish
Output: 1440
$4_P_2 \times 5_P_5 = 720$

Approach:

First, find the total no. of odd places and even places in the given word.

Total number of even places = floor(word length/2)

Total number of odd places = word length – total even places

Let’s consider the string “contribute” then there are 10 letters in the given word and there are 5 odd places, 5 even places, 4 vowels and 6 consonants.

Let us mark these positions as under:

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Now, 4 vowels can be placed at any of the five places, marked 1, 3, 5, 7, 9.

The number of ways of arranging the vowels = 5_P_4 = 5! = 120

Also, the 6 consonants can be arranged at the remaining 6 positions.

Number of ways of these arrangements = 6_P_6 = 6! = 720.

Total number of ways = (120 x 720) = 86400

Below is the implementation of the above approach:

C++

// C++ program to find the number of ways
// in which the characters of the word
// can be arranged such that the vowels
// occupy only the odd positions
#include <bits/stdc++.h>
using namespace std;
 
// Function to return the
// factorial of a number
int fact(int n)
{
    int f = 1;
    for (int i = 2; i <= n; i++) {
        f = f * i;
    }
 
    return f;
}
 
// calculating nPr
int npr(int n, int r)
{
    return fact(n) / fact(n - r);
}
 
// Function to find the number of ways
// in which the characters of the word
// can be arranged such that the vowels
// occupy only the odd positions
int countPermutations(string str)
{
    // Get total even positions
    int even = floor(str.length() / 2);
 
    // Get total odd positions
    int odd = str.length() - even;
 
    int ways = 0;
 
    // Store frequency of each character of
    // the string
    int freq[26] = { 0 };
    for (int i = 0; i < str.length(); i++) {
        ++freq[str[i] - 'a'];
    }
 
    // Count total number of vowels
    int nvowels
        = freq[0] + freq[4]
          + freq[8] + freq[14]
          + freq[20];
 
    // Count total number of consonants
    int nconsonants
        = str.length() - nvowels;
 
    // Calculate the total number of ways
    ways = npr(odd, nvowels) * npr(nconsonants, nconsonants);
 
    return ways;
}
 
// Driver code
int main()
{
    string str = "zambiatek";
 
    cout << countPermutations(str);
 
    return 0;
}

Java

// Java program to find the number of ways
// in which the characters of the word
// can be arranged such that the vowels
// occupy only the odd positions
class GFG{
// Function to return the
// 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 nPr
static int npr(int n, int r)
{
    return fact(n) / fact(n - r);
}
 
// Function to find the number of ways
// in which the characters of the word
// can be arranged such that the vowels
// occupy only the odd positions
static int countPermutations(String str)
{
    // Get total even positions
    int even = (int)Math.floor((double)(str.length() / 2));
 
    // Get total odd positions
    int odd = str.length() - even;
 
    int ways = 0;
 
    // Store frequency of each character of
    // the string
    int[] freq=new int[26];
    for (int i = 0; i < str.length(); i++) {
        freq[(int)(str.charAt(i)-'a')]++;
    }
 
    // Count total number of vowels
    int nvowels= freq[0] + freq[4]+ freq[8] 
                + freq[14]+ freq[20];
 
    // Count total number of consonants
    int nconsonants= str.length() - nvowels;
 
    // Calculate the total number of ways
    ways = npr(odd, nvowels) * npr(nconsonants, nconsonants);
 
    return ways;
}
 
// Driver code
public static void main(String[] args)
{
    String str = "zambiatek";
 
    System.out.println(countPermutations(str));
}
}
// This code is contributed by mits

Python3

# Python3 program to find the number 
# of ways in which the characters 
# of the word can be arranged such 
# that the vowels occupy only the 
# odd positions
import math
 
# Function to return the factorial
# of a number
def fact(n):
    f = 1;
    for i in range(2, n + 1):
        f = f * i;
 
    return f;
 
# calculating nPr
def npr(n, r):
    return fact(n) / fact(n - r);
 
# Function to find the number of 
# ways in which the characters of 
# the word can be arranged such 
# that the vowels occupy only the 
# odd positions
def countPermutations(str):
 
    # Get total even positions
    even = math.floor(len(str) / 2);
 
    # Get total odd positions
    odd = len(str) - even;
 
    ways = 0;
 
    # Store frequency of each 
    # character of the string
    freq = [0] * 26;
    for i in range(len(str)):
        freq[ord(str[i]) - ord('a')] += 1;
 
    # Count total number of vowels
    nvowels = (freq[0] + freq[4] + freq[8] +
               freq[14] + freq[20]);
 
    # Count total number of consonants
    nconsonants = len(str) - nvowels;
 
    # Calculate the total number of ways
    ways = (npr(odd, nvowels) *
            npr(nconsonants, nconsonants));
 
    return int(ways);
 
# Driver code
str = "zambiatek";
 
print(countPermutations(str));
     
# This code is contributed by mits

C#

// C# program to find the number of ways
// in which the characters of the word
// can be arranged such that the vowels
// occupy only the odd positions
using System;
class GFG{
// Function to return the
// 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 nPr
static int npr(int n, int r)
{
    return fact(n) / fact(n - r);
}
 
// Function to find the number of ways
// in which the characters of the word
// can be arranged such that the vowels
// occupy only the odd positions
static int countPermutations(String str)
{
    // Get total even positions
    int even = (int)Math.Floor((double)(str.Length / 2));
 
    // Get total odd positions
    int odd = str.Length - even;
 
    int ways = 0;
 
    // Store frequency of each character of
    // the string
    int[] freq=new int[26];
    for (int i = 0; i < str.Length; i++) {
        freq[(int)(str[i]-'a')]++;
    }
 
    // Count total number of vowels
    int nvowels= freq[0] + freq[4]+ freq[8] 
                + freq[14]+ freq[20];
 
    // Count total number of consonants
    int nconsonants= str.Length - nvowels;
 
    // Calculate the total number of ways
    ways = npr(odd, nvowels) * npr(nconsonants, nconsonants);
 
    return ways;
}
 
// Driver code
static void Main()
{
    String str = "zambiatek";
 
    Console.WriteLine(countPermutations(str));
}
}
// This code is contributed by mits

PHP

<?php
// PHP program to find the number 
// of ways in which the characters 
// of the word can be arranged such 
// that the vowels occupy only the 
// odd positions
 
// Function to return the
// factorial of a number
function fact($n)
{
    $f = 1;
    for ($i = 2; $i <= $n; $i++)
    {
        $f = $f * $i;
    }
 
    return $f;
}
 
// calculating nPr
function npr($n, $r)
{
    return fact($n) / fact($n - $r);
}
 
// Function to find the number 
// of $ways in which the characters 
// of the word can be arranged such 
// that the vowels occupy only the 
// odd positions
function countPermutations($str)
{
    // Get total even positions
    $even = floor(strlen($str)/ 2);
 
    // Get total odd positions
    $odd = strlen($str) - $even;
 
    $ways = 0;
 
    // Store $frequency of each 
    // character of the string
    $freq = array_fill(0, 26, 0);
    for ($i = 0; $i < strlen($str); $i++)
    {
        ++$freq[ord($str[$i]) - ord('a')];
    }
 
    // Count total number of vowels
    $nvowels= $freq[0] + $freq[4] + 
              $freq[8] + $freq[14] +
              $freq[20];
 
    // Count total number of consonants
    $nconsonants= strlen($str) - $nvowels;
 
    // Calculate the total number of ways
    $ways = npr($odd, $nvowels) *
            npr($nconsonants, $nconsonants);
 
    return $ways;
}
 
// Driver code
$str = "zambiatek";
 
echo countPermutations($str);
     
// This code is contributed by mits
?>

Javascript

<script>
// Javascript program to find the number
// of ways in which the characters
// of the word can be arranged such
// that the vowels occupy only the
// odd positions
 
// Function to return the
// factorial of a number
function fact(n)
{
    let f = 1;
    for (let i = 2; i <= n; i++)
    {
        f = f * i;
    }
 
    return f;
}
 
// calculating nPr
function npr(n, r)
{
    return fact(n) / fact(n - r);
}
 
// Function to find the number
// of ways in which the characters
// of the word can be arranged such
// that the vowels occupy only the
// odd positions
function countPermutations(str)
{
    // Get total even positions
    let even = Math.floor(str.length/ 2);
 
    // Get total odd positions
    let odd = str.length - even;
 
    let ways = 0;
 
    // Store frequency of each
    // character of the string
    let freq = new Array(26).fill(0);
    for (let i = 0; i < str.length; i++)
    {
        ++freq[str.charCodeAt(i) - 'a'.charCodeAt(0)];
    }
 
    // Count total number of vowels
    let nvowels= freq[0] + freq[4] +
            freq[8] + freq[14] +
            freq[20];
 
    // Count total number of consonants
    let nconsonants= str.length - nvowels;
 
    // Calculate the total number of ways
    ways = npr(odd, nvowels) *
            npr(nconsonants, nconsonants);
 
    return ways;
}
 
// Driver code
let str = "zambiatek";
 
document.write(countPermutations(str));
     
// This code is contributed by gfgking
</script>

Output

Complexity Analysis:

Time Complexity: O(n) where n is the length of the string
Auxiliary Space: O(26)

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