Problem Description
Given an integer n, count the number of strings of length n that consist only of vowels ('a', 'e', 'i', 'o', 'u') and are lexicographically sorted. A string is lexicographically sorted if every character is the same as or comes before the next character in alphabetical order.
Key Insights
- The string must be non-decreasing; once a vowel is chosen, further vowels cannot be lower in the alphabet.
- This is equivalent to finding combinations with repetition, where we choose n vowels from 5 with the constraint of sorted order.
- The problem can be solved using a combinatorial formula: the answer is "n+4 choose 4".
- A dynamic programming approach is also possible, but the combinatorial solution is more optimal and straightforward.
Space and Time Complexity
Time Complexity: O(n) (or O(1) if using the combinatorial formula since the loop runs only a constant number of times) Space Complexity: O(1)
Solution
The problem can be modeled as a combination with repetition: choosing n items from 5 types (vowels) where order does not matter. The number of ways to do this is given by the binomial coefficient C(n+5-1, 5-1), which simplifies to C(n+4, 4). To compute this result, we can iterate from 1 to 4 and multiply/divide accordingly. This approach avoids calculating factorials directly and prevents potential overflow issues for the given constraint (n ≤ 50).