algorithm - Why is the total number of possible substrings of a string n^2? -

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i read total number of substrings can formed given string n^2 don't understand how count this.

by substrings, mean, given string cat, substrings be:

c ca cat @ t

the total number of (nonempty) substrings n + c(n,2). leading n counts number of substrings of length 1 , c(n,2) counts number of substrings of length > 1 , equal number of ways choose 2 indices set of n. standard formula binomial coefficients yields c(n,2) = n*(n-1)/2. combining these 2 terms , simplifying gives total number (n^2 + n)/2. @rici in comments notes same c(n+1,2) makes sense if e.g. think in terms of python string slicing substrings of s can written in form s[i:j] 0 <= < j <= n (with j being 1 more final index). n = 3 works out (9 + 3)/2 = 6.

in sense of complexity theory number of substrings is o(n^2), might read somewhere.


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