How many substrings of different lengths (non-zero) can be found formed from a character string of length $n$?

How many substrings of different lengths (non-zero) can be found formed from a character string of length $n$?

(A) $n$, 

(B) $n^2$, 

(C) $\frac{n(n - 1)}{2}$, 

(D) $\frac{n(n + 1)}{2}$. 

(GATE 1994) 

Answer: (D) $\frac{n(n + 1)}{2}$.  

Explanation: 

The given string has length $n$. 

No. of substrings of length $n = 1$ 

No. of substrings of length $n - 1 = 2$ 

$\cdots \cdots \cdots \cdots \cdots \cdots$ 

$\cdots \cdots \cdots \cdots \cdots \cdots$ 

No. of substrings of length $1 = n$ 

Therefore, the total number of substrings 

$= 1 + 2 + \cdots + n$ 

$= \frac{n(n + 1)}{2}$.