Count sorted vowel strings: math approach

maths/count_sorted_vowel_strings.rb

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+# Challenge name: Count Sorted Vowel Strings
+#
+# 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.
+#
+#
+# Example 1:
+#
+# Input: n = 1
+# Output: 5
+# Explanation: The 5 sorted strings that consist of vowels only are ["a","e","i","o","u"].
+#
+# Example 2:
+#
+# Input: n = 2
+# Output: 15
+# Explanation: The 15 sorted strings that consist of vowels only are
+# ["aa","ae","ai","ao","au","ee","ei","eo","eu","ii","io","iu","oo","ou","uu"].
+# Note that "ea" is not a valid string since 'e' comes after 'a' in the alphabet.
+#
+# Example 3:
+#
+# Input: n = 33
+# Output: 66045
+
+
+#
+# Approach: Math
+#
+
+#
+# Intuition and Algorithm
+#
+# The problem is a variant of finding Combinations.
+# Mathematically, the problem can be described as,
+# given 5 vowels (let k = 5), we want to find the
+# number of combinations using only nn vowels. Also,
+# we can repeat each of those vowels multiple times.
+#
+
+# Complexity Analysis
+#
+# Time Complexity: O(1), as the approach runs in constant time.
+# Space Complexity: O(1), as the approach uses constant extra space.
+
+# @param {Integer} n
+# @return {Integer}
+def count_vowel_strings(n)
+  (n + 4) * (n + 3) * (n + 2) * (n + 1) / 24
+end
+
+n = 33
+puts count_vowel_strings(n)
+# Output: 66045
+
+n = 2
+puts count_vowel_strings(n)
+# Output: 15
+
+n = 1
+puts count_vowel_strings(n)
+# Output: 5