Find total number of non-empty substrings of a string with N characters. Here we use the word proper because we do not consider string itself as part of output.
Input : str = “abc”
Output : 6
Proper substrings are “a”, “b”, “c”, “ab”, “bc”, “abc”
Input : str = “abcd”
Output : 10
Proper substrings are “a”, “b”, “c”, “d”, “ab”, “bc”, “cd”, “abc”, “bcd” and “abcd”
Count of non-empty substrings is n*(n+1)/2
If we include empty string also as substring, the count becomes n*(n+1)/2 + 1
How does above formula work?
- Number of substrings of length one is n (We can choose any of the n characters)
- Number of substrings of length two is n-1 (We can choose any of the n-1 pairs formed by adjacent)
- Number of substrings of length three is n-2
(We can choose any of the n-2 triplets formed by adjacent)
- In general, mumber of substrings of length k is n-k+1 where 1 <= k <= n
Total number of substrings of all lengths from 1 to n =
n + (n-1) + (n-2) + (n-3) + … 2 + 1
= n * (n + 1)/2
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