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The number of substrings(of all lengths inclusive) that can be formed from character string of length n is

a)n

b)n^2

c)n(n-1)/2

d)(n(n+1)/2)+1
asked in Theory of Computation by (23 points) | 193 views
+1
let's take an example:

s = ABC

length 0 = 1

lenght 1 = {A , B , C} = 3

length 2 = {AB , BC } = 2

length 3 = {ABC} = 1

total string = length 0 + length 1 + length 2 + length  3 = 1 + 3 + 2 + 1 = 7

option D is the correct answer
0
For length 2,why have u not considered AC?
+1
@anji

AC is not  Substring of ABC   because it is separated by B , Substring is that in which we add another substring , in either left are Right side of Substring to get actual Substring.
0
+2

Substring is defined as "Zero or more Consecutive symbols taken"

What you must be confusing it with is "Subsequence" which is defined as "Zero or more Symbols left out (Consecutive or Not)"


Subword is defined as "One or more Consecutive symbols taken" (Subword is same as substring except for Zero length string)

1 Answer

+5 votes

The length of substrings are n or n-1 or n-2 or....or 2 or 1 or 0

let the string is abcd...xyz

no.of substrings of length 'n' = 1           ------  abcd....xyz

no.of substrings of length 'n-1' = 2       ------  abcd....xy , bcd....xyz

no.of substrings of length 'n-2' = 3       ------  abcd....x , bcd....xy , cdef.....xyz

.......

no.of substrings of length '2' = n-1      -----  ab,bc,cd,....vx,xy,yz

no.of substrings of length '1' = n         ------  a,b,c,d,....x,y,z

no.of substrings of length '0' = 1 ----> only empty string is possible

total substrings = 1+2+3+...+(n-1) + n + 1

                         = [ 1+2+3+...+(n-1) + n ] + 1

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

                         =  ∑n + 1

 

NOTE :-

Trivial substrings :- empty string ( which is length = 0 ) and the original string ( which is length = n ) are trivial strings.

                      ∴ No.of Trivial substrings are 2 for any non-empty string

Non- Trivial substrings :-  which are not Trivial substrings are called as Non- Trivial substrings.

                      ∴ No.of Non-Trivial substrings are ∑n - 1

Proper Substrings :- the substring which length is less than actual string

                      ∴ No.of Non-Trivial substrings are ∑n

Non-empty Substrings :- the substring which length is grater than 0

                      ∴ No.of Non-Empty substrings are ∑n

answered by Boss (49.9k points)
edited by
0
This formula works only, when all the alphabets in the given strings are distinct !

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