Day 29: Count goodStrings

day29/README.md

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+question of the day: https://codefights.com/interview/vzzkv5NHFMELWSwAS
+
+Given an integer len, count the number of different good strings that
+have a length of exactly len. A good string is a string for which the
+following conditions are true:
+
+1. A good string contains only lowercase English letters.
+2. Each character in a good string is unique.
+3. Exactly one character in a good string is lexicographically greater
+than the character that precedes it.
+
+**Example**
+
+For `len = 2`, the output should be
+`goodStringsCount(len) = 325`.
+
+If the first symbol is 'a', there are 25 possible good strings: "ab", "ac", ...  
+If the first symbol is 'b', there are 24 possible good strings: "bc", "bd", ...
+
+If the first symbol is 'z', there are 0 possible good strings because there is
+no character that is lexicographically greater.
+
+There are 25 + 24 + 23 + ... + 0 = 325 good strings that have a length of 2.
+
+For len = 1, the output should be goodStringsCount(len) = 0.
+
+The 3rd rule for good strings can't be true if there is only one character in the string.
+
+## Ideas
+
+Brute force: enumerate all possible permutations and check whether the
+constraints are satisfied. In this solution, 'aa', 'ab', etc. will be checked,
+but so will 'za', 'zb', 'zc', ..., 'zz', etc. which is a lot of wasted
+work. It also runs in `O(26**len)` which is exponential time which is horrible.
+
+How do we save work? Let's look at the constraints and factor them into the
+enumeration of permutations. We should never use upper case letters. We should
+not use duplicate letters, i.e. if we've placed the letter 'a' as we're
+constructing a good string, we should not place another letter 'a' in the same
+good string. And likewise, we shouldn't place letters into the good string
+if they are lexicographically less than the characters that precede it.
+
+len = 2  
+'ab', 'ac', 'ad', 'ae', ... 'az' -> 25 possible strings
+
+len = 3
+'abc', 'abd', 'abe', ... 'abz' -> 24 possible strings  
+'acd', 'ace', ... , 'acz' -> 23 possible strings  
+...  
+'ayz' -> 1 possible strings  
+'az' -> 0 possible strings
+
+'bcd' -> 23
+'bde' -> 22
+
+0 1 -> 25  
+1 2 -> 24  
+2 3 -> 23  
+3 4 -> 22  
+...
+25 26 -> 1  
+
+0 1 2 -> 24  
+1 2 3 -> 23  
+2 3 4 -> 22  
+3 4 5 -> 21  
+..
+23 24 25 -> 1
+24 25 _  -> 0
+
+## Code
+
+[Python](./goodStringscount.py)
+
+## Follow up
+
+I cheated today: http://blog.codefights.com/goodstringscount-solution/
+
+Be sure to read this and understand the combinatorics theory! I think it'd be
+a great exercise to break down a couple more combination questions. A lot of 
+companies seem to ask these kinds of questions, and it's a really useful math
+skill for calculating possibilities / probabilities / counting.
+
+

day29/goodStringsCount.py

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+from math import factorial
+
+def nChooseR(n, r):
+    return factorial(n) / (factorial(n-r) * factorial(r))
+
+def goodStringsCount(n):
+    return nChooseR(26, n) * (2**n - n - 1)
+
+def testGoodStringsCount():
+    assert goodStringsCount(1) == 0
+    assert goodStringsCount(2) == 325
+    assert goodStringsCount(3) == 10400
+
+def main():
+    testGoodStringsCount()
+
+if __name__ == "__main__":
+    main()