# Count N-length strings consisting only of vowels sorted lexicographically

Given an integer N, the task is to count all possible strings of length N consisting of vowels {a, e, i, o, u} that can be formed such that each string is sorted in lexicographical order.

Examples:

Input: N = 2
Output: 15
Explanation: The strings of length 2 which are sorted in lexicographical order are [“aa”, “ae”, “ai”, “ao”, “au”, “ee”, “ei”, “eo”, “eu”, “ii”, “io”, “iu”, “oo”, “ou”, “uu”].

Input: N = 1
Output: 5
Explanation: The strings of length 1 which are sorted in lexicographical order are [“a”, “e”, “i”, “o”, “u”].

Naive Approach: The simplest approach is to generate all possible strings of length N such that each string is sorted in lexicographical order. Print the count obtained after completing the steps.

Recursive Approach: Keep track of the vowel added to the string so that the next vowel added to the string is always Lexicographically greater.

## C++

 `// C++ program to illustrate Count of N-length strings` `// consisting only of vowels sorted lexicographically`   `#include ` `using` `namespace` `std;` `// to keep the string in lexicographically sorted order use` `// start index` `// to add the vowels starting the from that index` `int` `countstrings(``int` `n, ``int` `start)` `{` `    ``// base case: if string length is 0 add to the count` `    ``if` `(n == 0) {` `        ``return` `1;` `    ``}` `    ``int` `cnt = 0;` `    ``// if last character in string is 'e'` `    ``// add vowels starting from  'e'` `    ``// i.e    'e','i','o','u'` `    ``for` `(``int` `i = start; i < 5; i++) {` `        ``// decrease the length of string` `        ``cnt += countstrings(n - 1, i);` `    ``}` `    ``return` `cnt;` `}` `int` `countVowelStrings(``int` `n)` `{` `    ``//  char arr[5]={'a','e','i','o','u'};` `    ``// starting from index 0 add the vowels to strings` `    ``return` `countstrings(n, 0);` `}` `int` `main()` `{` `    ``int` `n = 2;` `    ``cout << countVowelStrings(n);` `    ``return` `0;` `}` `// This code is contributed by Deepak Chowdary`

## C

 `#include ` `// to keep the string in lexicographically sorted order use` `// start index` `// to add the vowels starting the from that index` `int` `countstrings(``int` `n, ``int` `start)` `{` `    ``// base case: if string length is 0 add to the count` `    ``if` `(n == 0) {` `        ``return` `1;` `    ``}` `    ``int` `cnt = 0;` `    ``// if last character in string is 'e'` `    ``// add vowels starting from  'e'` `    ``// i.e    'e','i','o','u'` `    ``for` `(``int` `i = start; i < 5; i++) {` `        ``// decrease the length of string` `        ``cnt += countstrings(n - 1, i);` `    ``}` `    ``return` `cnt;` `}` `int` `countVowelStrings(``int` `n)` `{` `    ``//  char arr[5]={'a','e','i','o','u'};` `    ``// starting from index 0 add the vowels to strings` `    ``return` `countstrings(n, 0);` `}` `//driver code` `int` `main() {`   `    ``int` `n = 2;` `    ``printf``(``"%d"``,countVowelStrings(n));` `    ``return` `0;` `}`   `// This code is contributed by thecodealpha.`

## Java

 `// Java program to illustrate Count of N-length strings` `// consisting only of vowels sorted lexicographically` `import` `java.util.*;` `public` `class` `Main` `{` `  `  `    ``// to keep the string in lexicographically sorted order use` `    ``// start index` `    ``// to add the vowels starting the from that index` `    ``static` `int` `countstrings(``int` `n, ``int` `start)` `    ``{` `       `  `        ``// base case: if string length is 0 add to the count` `        ``if` `(n == ``0``) {` `            ``return` `1``;` `        ``}` `        ``int` `cnt = ``0``;` `       `  `        ``// if last character in string is 'e'` `        ``// add vowels starting from  'e'` `        ``// i.e    'e','i','o','u'` `        ``for` `(``int` `i = start; i < ``5``; i++)` `        ``{` `           `  `            ``// decrease the length of string` `            ``cnt += countstrings(n - ``1``, i);` `        ``}` `        ``return` `cnt;` `    ``}` `    ``static` `int` `countVowelStrings(``int` `n)` `    ``{` `       `  `        ``//  char arr[5]={'a','e','i','o','u'};` `        ``// starting from index 0 add the vowels to strings` `        ``return` `countstrings(n, ``0``);` `    ``}` `    `  `    ``public` `static` `void` `main(String[] args) {` `        ``int` `n = ``2``;` `        ``System.out.print(countVowelStrings(n));` `    ``}` `}`   `// This code is contributed by divyesh072019.`

## Python3

 `# Python3 program to illustrate Count of N-length strings` `# consisting only of vowels sorted lexicographically`   `# to keep the string in lexicographically sorted order use` `# start index` `# to add the vowels starting the from that index` `def` `countstrings(n, start):` `  `  `    ``# base case: if string length is 0 add to the count` `    ``if` `n ``=``=` `0``:` `        ``return` `1` `    ``cnt ``=` `0` `    `  `    ``# if last character in string is 'e'` `    ``# add vowels starting from  'e'` `    ``# i.e    'e','i','o','u'` `    ``for` `i ``in` `range``(start, ``5``):` `      `  `        ``# decrease the length of string` `        ``cnt ``+``=` `countstrings(n ``-` `1``, i)` `    ``return` `cnt` `    `  `def` `countVowelStrings(n):` `  `  `    ``#  char arr[5]={'a','e','i','o','u'};` `    ``# starting from index 0 add the vowels to strings` `    ``return` `countstrings(n, ``0``)`   `n ``=` `2` `print``(countVowelStrings(n))`   `# This code is contributed by divyeshrabadiya07.`

## C#

 `// C# program to illustrate Count of N-length strings` `// consisting only of vowels sorted lexicographically` `using` `System;` `using` `System.Collections.Generic;` `class` `GFG {` `   `  `    ``// to keep the string in lexicographically sorted order use` `    ``// start index` `    ``// to add the vowels starting the from that index` `    ``static` `int` `countstrings(``int` `n, ``int` `start)` `    ``{` `      `  `        ``// base case: if string length is 0 add to the count` `        ``if` `(n == 0) {` `            ``return` `1;` `        ``}` `        ``int` `cnt = 0;` `      `  `        ``// if last character in string is 'e'` `        ``// add vowels starting from  'e'` `        ``// i.e    'e','i','o','u'` `        ``for` `(``int` `i = start; i < 5; i++) ` `        ``{` `          `  `            ``// decrease the length of string` `            ``cnt += countstrings(n - 1, i);` `        ``}` `        ``return` `cnt;` `    ``}` `    ``static` `int` `countVowelStrings(``int` `n)` `    ``{` `      `  `        ``//  char arr[5]={'a','e','i','o','u'};` `        ``// starting from index 0 add the vowels to strings` `        ``return` `countstrings(n, 0);` `    ``} `   `  ``static` `void` `Main() {` `    ``int` `n = 2;` `    ``Console.Write(countVowelStrings(n));` `  ``}` `}`   `// This code is contributed by decode2207.`

## Javascript

 ``

Output

`15`

Time Complexity: O(N!)
Auxiliary Space: O(1)

Efficient Approach: To optimize the above approach, the idea is to use Dynamic Programming. Below are some observations to solve the given problem:

• Count of lexicographically sorted strings of length 1 starting from characters a, e, i, o, and u is 1.
• Count of strings of length 2 that are in lexicographical order starting from characters a, e, i, o, and u is given by:
• The count of lexicographically sorted strings of length 2 starting from characters a is given by the count of the lexicographical strings of length 1 starting from character greater than or equal to a. Therefore, the count is 5.
• The count of lexicographically sorted strings of length 2 starting from characters e is given by the count of the lexicographical strings of length 1 starting from character greater than or equal to e. Therefore, the count is 4.
• The count of lexicographically sorted strings of length 2 starting from characters i is given by the count of the lexicographical strings of length 1 starting from character greater than or equal to i. Therefore, the count is 3.
• The count of lexicographically sorted strings of length 2 starting from characters o is given by the count of the lexicographical strings of length 1 starting from character greater than or equal to o. Therefore, the count is 2.
• The count of lexicographically sorted strings of length 2 starting from characters u is given by the count of the lexicographical strings of length 1 starting from character greater than or equal to u. Therefore, the count is 1.
• Therefore, the total count of strings length 2 is given by: 5 + 4 + 3 + 2 + 1 = 15.
• By observing the above pattern the count of strings of length N starting from each vowel character ch is given by the sum of the count of the lexicographical strings of length (N – 1) starting from character greater than or equal to ch.

Follow the steps below to solve the problem:

• Create a 2D array, dp[N + 1][6] where dp[i][j] represents the number of lexicographically sorted strings of length i that can be constructed using the first j vowels and initialize dp[1][1] with 1.
• Iterate over the first row using variable j, set dp[1][j] = dp[1][j – 1] + 1 as the string of length 1 are always sorted in lexicographically order.
• Traverse the 2D array dp[][] and update each dp state as dp[i][j] = dp[i][j – 1] + dp[i – 1][j], where dp[i][j – 1] will give the count of lexicographical string length N and dp[i – 1][j] will give the count of lexicographical string length (N – 1).
• After the above steps, print the value of dp[N][5] as the total count of resultant strings.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach` `#include ` `using` `namespace` `std;`   `// Function to count N-length strings` `// consisting of vowels only sorted` `// lexicographically` `int` `findNumberOfStrings(``int` `n)` `{` `    ``// Stores count of strings consisting` `    ``// of vowels sorted lexicographically` `    ``// of all possible lengths` `    ``vector > DP(n + 1,` `                            ``vector<``int``>(6));`   `    ``// Initialize DP[1][1]` `    ``DP[1][1] = 1;`   `    ``// Traverse the matrix row-wise` `    ``for` `(``int` `i = 1; i < n + 1; i++) {`   `        ``for` `(``int` `j = 1; j < 6; j++) {`   `            ``// Base Case` `            ``if` `(i == 1) {` `                ``DP[i][j] = DP[i][j - 1] + 1;` `            ``}`   `            ``else` `{` `                ``DP[i][j] = DP[i][j - 1]` `                           ``+ DP[i - 1][j];` `            ``}` `        ``}` `    ``}`   `    ``// Return the result` `    ``return` `DP[n][5];` `}`   `// Driver Code` `int` `main()` `{` `    ``int` `N = 2;`   `    ``// Function Call` `    ``cout << findNumberOfStrings(N);`   `    ``return` `0;` `}`

## C

 `#include ` `// Function to count N-length strings` `// consisting of vowels only sorted` `// lexicographically` `int` `findNumberOfStrings(``int` `n)` `{` `    ``// Stores count of strings consisting` `    ``// of vowels sorted lexicographically` `    ``// of all possible lengths` `    ``int` `DP[n + 1][6];` `    ``// Initialize DP[1][1]` `    ``DP[1][1] = 1;` `    ``// Traverse the matrix row-wise` `    ``for` `(``int` `i = 1; i < n + 1; i++) {`   `        ``for` `(``int` `j = 1; j < 6; j++) {`   `            ``// Base Case` `            ``if` `(i == 1)` `                ``DP[i][j] = DP[i][j - 1] + 1;` `            ``else` `                ``DP[i][j] = DP[i][j - 1] + DP[i - 1][j];` `        ``}` `    ``}` `    ``// Return the result` `    ``return` `DP[n][5];` `}` `// Driver Code` `int` `main() {`   `    ``int` `N = 2;` `    ``printf``(``"%d"``,findNumberOfStrings(N));` `    ``return` `0;` `}`   `// This code was contributed by Alok Khansali`

## Java

 `// Java program to implement` `// the above approach` `import` `java.util.*;` `class` `GFG{` `  `  `// Function to count N-length strings` `// consisting of vowels only sorted` `// lexicographically` `static` `int` `findNumberOfStrings(``int` `n)` `{` `  ``// Stores count of strings consisting` `  ``// of vowels sorted lexicographically` `  ``// of all possible lengths` `  ``int` `DP[][] = ``new` `int` `[n + ``1``][``6``];`   `  ``// Initialize DP[1][1]` `  ``DP[``1``][``1``] = ``1``;`   `  ``// Traverse the matrix row-wise` `  ``for` `(``int` `i = ``1``; i < n + ``1``; i++) ` `  ``{` `    ``for` `(``int` `j = ``1``; j < ``6``; j++) ` `    ``{` `      ``// Base Case` `      ``if` `(i == ``1``) ` `      ``{` `        ``DP[i][j] = DP[i][j - ``1``] + ``1``;` `      ``}`   `      ``else` `      ``{` `        ``DP[i][j] = DP[i][j - ``1``] + ` `                   ``DP[i - ``1``][j];` `      ``}` `    ``}` `  ``}`   `  ``// Return the result` `  ``return` `DP[n][``5``];` `}`   `// Driver Code` `public` `static` `void` `main(String[] args)` `{` `  ``int` `N = ``2``;`   `  ``// Function Call` `  ``System.out.print(findNumberOfStrings(N));` `}` `}`   `// This code is contributed by sanjoy_62`

## Python3

 `# Python3 program for the ` `# above approach`   `# Function to count N-length ` `# strings consisting of vowels ` `# only sorted lexicographically` `def` `findNumberOfStrings(n):` `  `  `    ``# Stores count of strings ` `    ``# consisting of vowels ` `    ``# sorted lexicographically` `    ``# of all possible lengths` `    ``DP ``=` `[[``0` `for` `i ``in` `range``(``6``)] ` `             ``for` `i ``in` `range``(n ``+` `1``)]`   `    ``# Initialize DP[1][1]` `    ``DP[``1``][``1``] ``=` `1`   `    ``# Traverse the matrix row-wise` `    ``for` `i ``in` `range``(``1``, n ``+` `1``):` `        ``for` `j ``in` `range``(``1``, ``6``):`   `            ``#Base Case` `            ``if` `(i ``=``=` `1``):` `                ``DP[i][j] ``=` `DP[i][j ``-` `1``] ``+` `1` `            ``else``:` `                ``DP[i][j] ``=` `DP[i][j ``-` `1``]``+` `DP[i ``-` `1``][j]`   `    ``# Return the result` `    ``return` `DP[n][``5``]`   `# Driver Code` `if` `__name__ ``=``=` `'__main__'``:` `  `  `    ``N ``=` `2`   `    ``# Function Call` `    ``print``(findNumberOfStrings(N))`   `# This code is contributed by Mohit Kumar 29`

## C#

 `// C# program to implement` `// the above approach` `using` `System;` `class` `GFG{` `  `  `// Function to count N-length strings` `// consisting of vowels only sorted` `// lexicographically` `static` `int` `findNumberOfStrings(``int` `n)` `{` `  ``// Stores count of strings consisting` `  ``// of vowels sorted lexicographically` `  ``// of all possible lengths` `  ``int``[,] DP = ``new` `int` `[n + 1, 6];`   `  ``// Initialize DP[1][1]` `  ``DP[1, 1] = 1;`   `  ``// Traverse the matrix row-wise` `  ``for` `(``int` `i = 1; i < n + 1; i++) ` `  ``{` `    ``for` `(``int` `j = 1; j < 6; j++) ` `    ``{` `      ``// Base Case` `      ``if` `(i == 1) ` `      ``{` `        ``DP[i, j] = DP[i, j - 1] + 1;` `      ``}`   `      ``else` `      ``{` `        ``DP[i, j] = DP[i, j - 1] + ` `                   ``DP[i - 1, j];` `      ``}` `    ``}` `  ``}`   `  ``// Return the result` `  ``return` `DP[n, 5];` `}`   `// Driver Code` `public` `static` `void` `Main(``string``[] args)` `{` `  ``int` `N = 2;`   `  ``// Function Call` `  ``Console.Write(findNumberOfStrings(N));` `}` `}`   `// This code is contributed by Chitranayal`

## Javascript

 ``

Output

`15`

Time Complexity: O(N*5)
Auxiliary Space: O(N*5)

Efficient Approach: The above approach can further be simplified to linear time and constant space.

Here are some of the observations for different lengths of strings:-

It is seen that for each value of N, number of strings possible is dependent on the previous value of N (N-1).

Value of any column in the Nth row is the sum of all columns in the (N-1)th row, starting from right hand side upto that column number.

## C++

 `#include ` `using` `namespace` `std;`   `// Function to count N-length strings` `// consisting of vowels only sorted` `// lexicographically` `int` `findNumberOfStrings(``int` `N)` `{` `    ``// Initializing vector to store count of strings.` `    ``vector<``int``> counts(5, 1);`   `    ``for` `(``int` `i = 2; i <= N; i++) {` `        ``for` `(``int` `j = 3; j >= 0; j--)` `            ``counts[j] += counts[j + 1];` `    ``}`   `    ``int` `ans = 0;`   `    ``// Summing up the total number of combinations.` `    ``for` `(``auto` `c : counts)` `        ``ans += c;`   `    ``// Return the result` `    ``return` `ans;` `}`   `// Driver Code` `int` `main()` `{` `    ``int` `N = 2;`   `    ``// Function Call` `    ``cout << findNumberOfStrings(N);`   `    ``return` `0;` `}`   `// This code is contributed by Sarvesh Roshan.`

## C

 `#include ` `// Function to count N-length strings` `// consisting of vowels only sorted` `// lexicographically` `int` `findNumberOfStrings(``int` `N)` `{` `    ``// Initializing vector to store count of strings.` `    ``int` `counts[5];` `    ``for``(``int` `i = 0; i < 5;i++)` `      ``counts[i] = 1;` `    ``for` `(``int` `i = 2; i <= N; i++) {` `        ``for` `(``int` `j = 3; j >= 0; j--)` `            ``counts[j] += counts[j + 1];` `    ``}` `    ``int` `ans = 0;` `    ``// Summing up the total number of combinations.` `    ``for` `(``int` `c = 0; c < 5; c++)` `        ``ans += counts;` `    ``// Return the result` `    ``return` `ans;` `}` `// Driver Code` `int` `main()` `{` `    ``int` `N = 2;` `    ``printf``(``"%d"``,findNumberOfStrings(N));` `    ``return` `0;` `}` `//This code was contributed by Alok Khansali`

## Java

 `// Java program for the above approach` `import` `java.util.*;` `public` `class` `Main` `{` `    ``// Function to count N-length strings` `    ``// consisting of vowels only sorted` `    ``// lexicographically` `    ``static` `int` `findNumberOfStrings(``int` `N)` `    ``{` `        ``// Initializing vector to store count of strings.` `        ``Vector counts = ``new` `Vector();` `        ``for``(``int` `i = ``0``; i < ``5``; i++)` `        ``{` `            ``counts.add(``1``);` `        ``}` `     `  `        ``for` `(``int` `i = ``2``; i <= N; i++) {` `            ``for` `(``int` `j = ``3``; j >= ``0``; j--)` `                ``counts.set(j, counts.get(j) + counts.get(j + ``1``));` `        ``}` `     `  `        ``int` `ans = ``0``;` `     `  `        ``// Summing up the total number of combinations.` `        ``for``(Integer c : counts)` `            ``ans += c;` `     `  `        ``// Return the result` `        ``return` `ans;` `    ``}` `    `  `    ``public` `static` `void` `main(String[] args) {` `        ``int` `N = ``2``;` ` `  `        ``// Function Call` `        ``System.out.print(findNumberOfStrings(N));` `    ``}` `}`   `// This code is contributed by mukesh07.`

## Python3

 `# Python3 program for the above approach`   `# Function to count N-length strings` `# consisting of vowels only sorted` `# lexicographically` `def` `findNumberOfStrings(N):` `    ``# Initializing vector to store count of strings.` `    ``counts ``=` `[]` `    ``for` `i ``in` `range``(``5``):` `        ``counts.append(``1``)` `  `  `    ``for` `i ``in` `range``(``2``, N ``+` `1``):` `        ``for` `j ``in` `range``(``3``, ``-``1``, ``-``1``):` `            ``counts[j] ``+``=` `counts[j ``+` `1``]` `  `  `    ``ans ``=` `0` `  `  `    ``# Summing up the total number of combinations.` `    ``for` `c ``in` `counts:` `        ``ans ``+``=` `c` `  `  `    ``# Return the result` `    ``return` `ans`   `N ``=` `2` `  `  `# Function Call` `print``(findNumberOfStrings(N))`   `# This code is contributed by decode2207.`

## C#

 `// C# program to implement above approach` `using` `System;` `using` `System.Collections;` `using` `System.Collections.Generic;`   `class` `GFG` `{` `  ``// Function to count N-length strings` `  ``// consisting of vowels only sorted` `  ``// lexicographically` `  ``static` `int` `findNumberOfStrings(``int` `N)` `  ``{` `    ``// Initializing vector to store count of strings.` `    ``List<``int``> counts = ``new` `List<``int``>();` `    ``for``(``int` `i = 0 ; i < 5 ; i++)` `    ``{` `      ``counts.Add(1);` `    ``}`   `    ``for` `(``int` `i = 2 ; i <= N ; i++) {` `      ``for` `(``int` `j = 3 ; j >= 0 ; j--){` `        ``counts[j] = counts[j] + counts[j+1];` `      ``}` `    ``}`   `    ``int` `ans = 0;`   `    ``// Summing up the total number of combinations.` `    ``foreach` `(``int` `c ``in` `counts)` `      ``ans += c;`   `    ``// Return the result` `    ``return` `ans;` `  ``}`   `  ``// Driver code` `  ``public` `static` `void` `Main(``string``[] args){`   `    ``int` `N = 2;`   `    ``// Function Call` `    ``Console.Write(findNumberOfStrings(N));`   `  ``}` `}`   `// This code is contributed by subhamgoyal2014.`

## Javascript

 ``

Output

`15`

Time Complexity: O(5*N)
Space Complexity: O(1)

Efficient Approach: The same idea of the above dp approach can be implemented in constant time and constant space.

## C++

 `// C++ program for the above approach` `#include ` `using` `namespace` `std;`   `// Function to count N-length strings` `// consisting of vowels only sorted` `// lexicographically` `int` `findNumberOfStrings(``int` `n)` `{` `    ``return` `(n+1)*(n+2)*(n+3)*(n+4)/24;` `}`   `// Driver Code` `int` `main()` `{` `    ``int` `N = 2;`   `    ``// Function Call` `    ``cout << findNumberOfStrings(N);`   `    ``return` `0;` `}` `// This code is contributed by Kartik Singh`

## C

 `#include ` `int` `findNumberOfStrings(``int` `n)` `{` `    ``return` `(n+1)*(n+2)*(n+3)*(n+4)/24;` `}` `// Driver Code` `int` `main()` `{` `    ``int` `N = 2;` `    ``printf``(``"%d"``, findNumberOfStrings(N));` `    ``return` `0;` `}` `//This code has been added by Alok Khansali`

## Java

 `// Java program to implement` `// the above approach` `import` `java.util.*;` `class` `GFG{` `  `  `// Function to count N-length strings` `// consisting of vowels only sorted` `// lexicographically` `static` `int` `findNumberOfStrings(``int` `n)` `{` ` ``return` `(n+``1``)*(n+``2``)*(n+``3``)*(n+``4``)/``24``;` `}`   `// Driver Code` `public` `static` `void` `main(String[] args)` `{` `  ``int` `N = ``2``;`   `  ``// Function Call` `  ``System.out.print(findNumberOfStrings(N));` `}` `}`   `// This code is contributed by Kartik Singh`

## Python

 `# Python3 program for the ` `# above approach`   `# Function to count N-length ` `# strings consisting of vowels ` `# only sorted lexicographically` `def` `findNumberOfStrings(n):` `    ``return` `int``((n``+``1``)``*``(n``+``2``)``*``(n``+``3``)``*``(n``+``4``)``/``24``)`   `# Driver Code` `if` `__name__ ``=``=` `'__main__'``:` `  `  `    ``N ``=` `2`   `    ``# Function Call` `    ``print``(findNumberOfStrings(N))`   `# This code is contributed by Kartik Singh`

## C#

 `// C# program to implement` `// the above approach` `using` `System;` `class` `GFG{` `  `  `// Function to count N-length strings` `// consisting of vowels only sorted` `// lexicographically` `static` `int` `findNumberOfStrings(``int` `n)` `{` `  ``return` `(n+1)*(n+2)*(n+3)*(n+4)/24;` `}`   `// Driver Code` `public` `static` `void` `Main(``string``[] args)` `{` `  ``int` `N = 2;`   `  ``// Function Call` `  ``Console.Write(findNumberOfStrings(N));` `}` `}`   `// This code is contributed by Kartik Singh`

## Javascript

 ``

Output

`15`

Time Complexity: O(1)

Auxiliary Space: O(1)

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