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+11 votes

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

  1. $n$
  2. $n^2$
  3. $2^n$
  4. $\frac{n(n+1)}{2}$


asked in Combinatory by Veteran (69k points)
retagged by | 1.9k views
Anyone please confirm ans ... I am getting option A (n) as correct answer as question is asking for different length substrings.

Aswer will be D  (asuming all characters in string are distinct)

Take an example String "abc"

Substrings of length 1 = {a,b,c}

Substrings of length 2 = {ab,bc}

Substrings of length 3 = {abc}

Total Substrings of non zero length will be 3+2+1 = 6

However Answer will be A, if all characters are same

consider "aaa"

Substrings of length 1 = {a}

Substrings of length 2 = {aa}

Substrings of length 3 = {aaa}

So answer can be vary from n to $\frac{n(n+1)}{2}$ if the distinct keyword is not given!!


^ what do you mean by - strings of different lengths (non-zero)  ??



strings of different lengths (non-zero) 

means some string are of length 1

Some string lengh 2

................some string length n


it doesnot mean each string of different length

Okay, thanks:)
why is substring different from subsets?
So some substrings will be of length 1, some of length 2 and so on. Since it is mentioned how many substrings of different length, from each set we take one substring i.e. one substring of length 1, one substring of length 2 and so on till one substring of length n. Thus, total substrings of different length possible are n. What is wrong with this reasoning?

2 Answers

+20 votes
Best answer
Assuming in the string of length $n$ provided, all alphabets are distinct.

No. of strings of length $1 = n$

No. of strings of length $2=\left(n-1\right)$

No. of strings of length $3=\left(n-2\right)$
No. of strings of length $n = 1$

$\begin{align} \text{Hence, total no. of strings} &= n + (n -1) + (n - 2) + (n - 2) +\cdots+ 1\\&= \frac{n(n+1)}{2}\end{align}$
answered by Veteran (54k points)
edited by

Lemme know what they actually mean by "different length"? ..according to your answer you are assuming like

# total strings of length1+ # total strings of length2 ...+ #total strings of length n ....

But there are n strings of length 1 (same langth) ,n-1 of length 2 (same length) ....1 string of length n answer can be 'n' in that way...

So what they actually mean..


Let string be {a,b,c}

subs of length 2 are ab,bc,ac .how is it then following no of strings of length 2 = n-1.plz clarify


@Gate Mm

if abc is a string, then length 2 sub-strings will be ab,bc only.
Because substring is the part of string. So is not part of the string.
How to know if all alphabets are distinct? If they are same then we have the option (a) as the answer
how ac is not part of substring?
–2 votes

The correct answer is (D) n⨉(n+1)/2

answered by Veteran (16.5k points)
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