Count anagrams having first character as a consonant and no pair of consonants or vowels placed adjacently

Given a string S of length N, the task is to count the number of anagrams of S whose first character is a consonant and no pair of consonants or vowels are adjacent to each other.

Examples:

Input: S = “GADO”
Output: 4
Explanation:
The anagrams of string S satisfying the given conditions are GADO, GODA, DOGA, DAGO.
Therefore, the total number of such anagrams is 4.

Input: S = “AABCY”
Output: 6
Explanation:
The anagrams of the string S satisfying the given conditions are BACAY, BAYAC, CABAY, CAYAB, YABAC, YACAB.
Therefore, the total number of such anagrams is 6.

Naive Approach: The simplest approach is to generate all possible anagrams of the given string and count those anagrams that satisfy the given condition. Finally, print the count obtained.
Time Complexity: O(N!*N)
Auxiliary Space: O(1)

Efficient Approach: The above approach can also be optimized based on the following observations:

• Strings that have an equal number of consonants and vowels satisfy the given condition.
• Strings having one more consonant than vowel also satisfy the given condition.
• Apart from these two conditions, the count of possible anagrams will always be 0.
• Now, the problem can be solved by using a combinatorial formula. Consider there are C₁, C₂…, C_N consonants and V₁, V₂, …, V_N vowels in the string S and  and \sum C denote the total number of consonants and vowels respectively, then the answer would be:

where, 
C_i is the count of i^th consonant.
_Vi is the count of i^th vowel.

Follow the steps below to solve the problem:

• Initialize a variable, say answer, to store the total count of anagrams.
• Store the frequency of each character of the string S in a HashMap count.
• Store the number of vowels and consonants in S in variables V and C respectively.
• If the value of V is not equal to C or C is not equal to (V + 1), then print 0. Otherwise, performing the following steps:
  • Initialize denominator as 1.
  • Traverse the string S using the variable i and update the denominator as denominator*((count[S[i]])!).
  • Initialize numerator to V!*C!, and update the value of answer as numerator/denominator.
• After completing the above steps, print the value of the answer as the result.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>

#define ll long long
#define mod 1000000007
#define N 1000001
using namespace std;

// Function to compute factorials till N
void Precomputefact(unordered_map<ll, ll>& fac)
{
    ll ans = 1;

    // Iterate in the range [1, N]
    for (ll i = 1; i <= N; i++) {

        // Update ans to ans*i
        ans = (ans * i) % mod;

        // Store the value of ans
        // in fac[i]
        fac[i] = ans;
    }
    return;
}

// Function to check whether the
// current character is a vowel or not
bool isVowel(char a)
{
    if (a == 'A' || a == 'E' || a == 'I' || a == 'O'
        || a == 'U')
        return true;
    else
        return false;
}

// Function to count the number of
// anagrams of S satisfying the
// given condition
void countAnagrams(string s, int n)
{
    // Store the factorials upto N
    unordered_map<ll, ll> fac;

    // Function Call to generate
    // all factorials upto n
    Precomputefact(fac);

    // Create a hashmap to store
    // frequencies of all characters
    unordered_map<char, ll> count;

    // Store the count of
    // vowels and consonants
    int vo = 0, co = 0;

    // Iterate through all
    // characters in the string
    for (int i = 0; i < n; i++) {

        // Update the frequency
        // of current character
        count[s[i]]++;

        // Check if the character
        // is vowel or consonant
        if (isVowel(s[i]))
            vo++;
        else
            co++;
    }

    // Check if ΣC==ΣV+1 or ΣC==ΣV
    if ((co == vo + 1) || (co == vo)) {

        // Store the denominator
        ll deno = 1;

        // Calculate the denominator
        // of the expression
        for (auto c : count) {

            // Multiply denominator by factorial
            // of counts of all letters
            deno = (deno * fac[c.second]) % mod;
        }

        // Store the numerator
        ll nume = fac[co] % mod;
        nume = (nume * fac[vo]) % mod;

        // Store the answer by dividing
        // numerator by denominator
        ll ans = nume / deno;

        // Print the answer
        cout << ans;
    }

    // Otherwise, print 0
    else {
        cout << 0;
    }
}

// Driver Code
int main()
{
    string S = "GADO";
    int l = S.size();
    countAnagrams(S, l);

    return 0;
}

Java

import java.util.*;

class Main {
  static int mod = 1000000007;
  static int N = 1000001;
  static HashMap<Integer, Long> fac
    = new HashMap<Integer, Long>();

  // Function to compute factorials till N
  static void precomputeFact()
  {
    long ans = 1;

    // Iterate in the range [1, N]
    for (int i = 1; i <= N; i++)
    {

      // Update ans to ans*i
      ans = (ans * i) % mod;

      // Store the value of ans in fac[i]
      fac.put(i, ans);
    }
  }

  // Function to check whether the current character is a
  // vowel or not
  static boolean isVowel(char a)
  {
    return a == 'A' || a == 'E' || a == 'I' || a == 'O'
      || a == 'U';
  }

  // Function to count the number of anagrams of S
  // satisfying the given condition
  static void countAnagrams(String s, int n)
  {
    // Store the factorials upto N
    // Function Call to generate all factorials upto n
    precomputeFact();

    // Create a hashmap to store frequencies of all
    // characters
    HashMap<Character, Integer> count = new HashMap<>();

    // Store the count of vowels and consonants
    int vo = 0;
    int co = 0;

    // Iterate through all characters in the string
    for (int i = 0; i < n; i++) {
      // Update the frequency of current character
      if (count.containsKey(s.charAt(i))) {
        count.put(s.charAt(i),
                  count.get(s.charAt(i)) + 1);
      }
      else {
        count.put(s.charAt(i), 1);
      }

      // Check if the character is vowel or consonant
      if (isVowel(s.charAt(i))) {
        vo++;
      }
      else {
        co++;
      }
    }

    if (co == vo + 1 || co == vo) {
      // Store the denominator of the expression
      long deno = 1;

      // Calculate the denominator of the expression
      for (var item : count.entrySet()) {
        // Multiply denominator by factorial of
        // counts of all letters
        deno = (deno * fac.get(item.getValue()))
          % mod;
      }

      // Store the numerator
      long nume = fac.get(co) % mod;
      nume = (nume * fac.get(vo)) % mod;

      // Store the answer by dividing numerator by
      // denominator
      long ans = nume / deno;

      // Print the answer
      System.out.println(ans);
    }
    else {
      System.out.println(0);
    }
  }

  public static void main(String[] args)
  {
    String S = "GADO";
    int l = S.length();
    countAnagrams(S, l);
  }
}

// This code is contributed by phasing17.

Python3

# Python 3 program for the above approach
#include <bits/stdc++.h>

mod = 1000000007
N = 1000001

fac = {}

# Function to compute factorials till N
def Precomputefact():
    global fac
    ans = 1

    # Iterate in the range [1, N]
    for i in range(1,N+1,1):
        # Update ans to ans*i
        ans = (ans * i) % mod

        # Store the value of ans
        # in fac[i]
        fac[i] = ans

    return

# Function to check whether the
# current character is a vowel or not
def isVowel(a):
    if (a == 'A' or a == 'E' or a == 'I' or a == 'O' or a == 'U'):
        return True
    else:
        return False

# Function to count the number of
# anagrams of S satisfying the
# given condition
def countAnagrams(s,n):
    # Store the factorials upto N
    global fac

    # Function Call to generate
    # all factorials upto n
    Precomputefact()

    # Create a hashmap to store
    # frequencies of all characters
    count = {}

    # Store the count of
    # vowels and consonants
    vo = 0
    co = 0

    # Iterate through all
    # characters in the string
    for i in range(n):
        # Update the frequency
        # of current character
        if s[i] in count:
            count[s[i]] += 1
        else:
            count[s[i]] = 1

        # Check if the character
        # is vowel or consonant
        if (isVowel(s[i])):
            vo += 1
        else:
            co += 1

    # Check if ΣC==ΣV+1 or ΣC==ΣV
    if ((co == vo + 1) or (co == vo)):
        # Store the denominator
        deno = 1

        # Calculate the denominator
        # of the expression
        for key,value in count.items():
            # Multiply denominator by factorial
            # of counts of all letters
            deno = (deno * fac[value]) % mod

        # Store the numerator
        nume = fac[co] % mod
        nume = (nume * fac[vo]) % mod

        # Store the answer by dividing
        # numerator by denominator
        ans = nume // deno

        # Print the answer
        print(ans)

    # Otherwise, print 0
    else:
        print(0)

# Driver Code
if __name__ == '__main__':
    S = "GADO"
    l = len(S)
    countAnagrams(S, l)

    # This code is contributed by ipg2016107.

C#

// C# program for the above approach
using System;
using System.Collections.Generic;
using System.Linq;

class GFG {
  static int mod = 1000000007;
  static int N = 1000001;
  static Dictionary<int, long> fac
    = new Dictionary<int, long>();

  // Function to compute factorials till N
  static void Precomputefact()
  {
    long ans = 1;

    // Iterate in the range [1, N]
    for (int i = 1; i <= N; i++) {
      // Update ans to ans*i
      ans = (ans * i) % mod;

      // Store the value of ans
      // in fac[i]
      fac[i] = ans;
    }
  }

  // Function to check whether the
  // current character is a vowel or not
  static bool IsVowel(char a)
  {
    return (a == 'A' || a == 'E' || a == 'I' || a == 'O'
            || a == 'U');
  }

  // Function to count the number of
  // anagrams of S satisfying the
  // given condition
  static void CountAnagrams(string s, int n)
  {
    // Store the factorials upto N
    // Function Call to generate
    // all factorials upto n
    Precomputefact();

    // Create a hashmap to store
    // frequencies of all characters
    Dictionary<char, int> count
      = new Dictionary<char, int>();

    // Store the count of
    // vowels and consonants
    int vo = 0;
    int co = 0;

    // Iterate through all
    // characters in the string
    for (int i = 0; i < n; i++) {
      // Update the frequency
      // of current character
      if (count.ContainsKey(s[i])) {
        count[s[i]]++;
      }
      else {
        count[s[i]] = 1;
      }

      // Check if the character
      // is vowel or consonant
      if (IsVowel(s[i])) {
        vo++;
      }
      else {
        co++;
      }
    }

    // Check if ΣC==ΣV+1 or ΣC==ΣV
    if (co == vo + 1 || co == vo) {
      // Store the denominator
      long deno = 1;

      // Calculate the denominator
      // of the expression
      foreach(var item in count)
      {
        // Multiply denominator by factorial
        // of counts of all letters
        deno = (deno * fac[item.Value]) % mod;
      }

      // Store the numerator
      long nume = fac[co] % mod;
      nume = (nume * fac[vo]) % mod;

      // Store the answer by dividing
      // numerator by denominator
      long ans = nume / deno;

      // Print the answer
      Console.WriteLine(ans);
    }
    else
      Console.WriteLine(0);
  }

  public static void Main(string[] args)
  {

    string S = "GADO";
    int l = S.Length;
    CountAnagrams(S, l);
  }
}

// This code is contributed by phasing17.

Javascript

// JS program to implement the approach

const mod = 1000000007;
const N = 1000001;

let fac = {};

// Function to compute factorials till N
function Precomputefact() {
  let ans = 1;

  // Iterate in the range [1, N]
  for (let i = 1; i <= N; i++) {
    // Update ans to ans*i
    ans = (ans * i) % mod;

    // Store the value of ans in fac[i]
    fac[i] = ans;
  }

  return;
}

// Function to check whether the current character is a vowel or not
function isVowel(a) {
  if (a == "A" || a == "E" || a == "I" || a == "O" || a == "U") {
    return true;
  } else {
    return false;
  }
}

// Function to count the number of anagrams of S satisfying the given condition
function countAnagrams(s, n) {
  // Store the factorials upto N

  // Function Call to generate all factorials upto n
  Precomputefact();

  // Create a hashmap to store frequencies of all characters
  let count = {};

  // Store the count of vowels and consonants
  let vo = 0;
  let co = 0;

  // Iterate through all characters in the string
  for (let i = 0; i < n; i++) {
    // Update the frequency of current character
    if (s[i] in count) {
      count[s[i]] += 1;
    } else {
      count[s[i]] = 1;
    }

    // Check if the character is vowel or consonant
    if (isVowel(s[i])) {
      vo += 1;
    } else {
      co += 1;
    }
  }

  // Check if ΣC==ΣV+1 or ΣC==ΣV
  if (co == vo + 1 || co == vo) {
    // Store the denominator of the expression
    let deno = 1;

    // Calculate the denominator of the expression
    for (let key of Object.keys(count)) {
      // Multiply denominator by factorial of counts of all letters
      deno = (deno * fac[count[key]]) % mod;
    }

    // Store the numerator
    let nume = fac[co] % mod;
    nume = (nume * fac[vo]) % mod;

    // Store the answer by dividing numerator by denominator
    let ans = Math.floor(nume / deno);

    // Print the answer
    console.log(ans);
  } else {
    // Otherwise, print 0
    console.log(0);
  }
}

// Driver Code
let S = "GADO";
let l = S.length;

countAnagrams(S, l);


// This code is contributed by phasing17

Output:

4

Time Complexity: O(N)
Auxiliary Space: O(N)