Count Sorted Vowel Strings

Count Sorted Vowel Strings - Problem

Given an integer n, return the number of strings of length n that consist only of vowels (a, e, i, o, u) and are lexicographically sorted.

A string s is lexicographically sorted if for all valid i, s[i] is the same as or comes before s[i+1] in the alphabet.

Input & Output

Example 1 — Basic Case

$ Input: n = 1

› Output: 5

💡 Note: All single vowel strings are valid: 'a', 'e', 'i', 'o', 'u' are all lexicographically sorted by themselves

Example 2 — Small Length

$ Input: n = 2

› Output: 15

💡 Note: Valid strings include 'aa', 'ae', 'ai', 'ao', 'au', 'ee', 'ei', 'eo', 'eu', 'ii', 'io', 'iu', 'oo', 'ou', 'uu'. Invalid would be 'ea', 'ia', etc.

Example 3 — Larger Case

$ Input: n = 33

› Output: 66045

💡 Note: For n=33, using the mathematical formula C(33+4, 4) = C(37, 4) = 66045 valid sorted vowel strings

Constraints

1 ≤ n ≤ 50

Visualization

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Asked in

G Google 15 a Amazon 12

The key insight is recognizing this as a combinatorics problem equivalent to choosing vowels with repetition where order doesn't matter. The optimal approach uses the mathematical formula C(n+4, 4) for constant time solution. Time: O(1), Space: O(1)

Common Approaches

✓ Prefix Sum

⏱️ Time: N/A Space: N/A

Combinatorics Formula

⏱️ Time: O(1) Space: O(1)

This problem is equivalent to finding the number of ways to choose n items from 5 vowels where repetition is allowed and order doesn't matter (since we want sorted strings). This is the classic 'stars and bars' combinatorics problem.

Dynamic Programming

⏱️ Time: O(n) Space: O(1)

Build up the solution by tracking how many valid strings of each length can end with each vowel. For each position, we can only add vowels that are greater than or equal to the current last vowel.

Recursive Generation

⏱️ Time: O(5^n) Space: O(n)

Recursively build all possible strings of length n using vowels, then check if each string is lexicographically sorted. This explores the entire solution space systematically.

Algorithm Steps — Algorithm Steps

Code -

solution.c — C

Time & Space Complexity

Time Complexity

⏱️

✓ Linear Growth

Space Complexity

✓ Linear Space

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Medium Frequency

~15 min Avg. Time

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Smart Actions

💡 Explanation

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