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Let the i/p alphabet = {0,1} for the following questions :

$\color{red}{Q) \mathbf{About\; String \;Lengths : }}$

$\color{green}{i) \text{ Length of the strings exactly = n  :  }}\color{blue}{\text{  n+2}}$

$\color{green}{ii) \text{ Length of the strings atmost = n  :  }}\color{blue}{\text{  n+2}}$

$\color{green}{iii) \text{ Length of the strings atleast = n  :  }}\color{blue}{\text{  n+1}}$

$\color{green}{iv) \text{ Length of the strings divisible by n  :  }}\color{blue}{\text{  n}}$

$\color{green}{v) \text{ Length of the strings atleast = m & atmost = n  :  }}\color{blue}{\text{  n+2}}\color{magenta}{\text{ [ note:- n ≥ m ]}}$

 

$\color{red}{Q) \mathbf{About\; Prefix\; Lengths : }}$

$\color{green}{i) \text{ Length of the prefix  = n  :  }}\color{blue}{\text{  n+2}}$

$\color{green}{ii) \text{ Length of the prefix  = m & total length of the string exactly = n :  }}\color{blue}{\text{ n+2}}\color{magenta}{\text{ [ note:- n ≥ m ]}}$

$\color{green}{iii) \text{ Length of the prefix  = m & total length of the string atmost = n :  }}\color{blue}{\text{ n+2}}\color{magenta}{\text{ [ note:- n ≥ m ]}}$

$\color{green}{iv) \text{ Length of the prefix  = m & total length of the string atleast = n :  }}\color{blue}{\text{  max(m,n)+2}}$

$\color{green}{v) \text{ Length of the prefix  = m & total length of the string divisible by n :  }}\color{blue}{\text{ m+n+1}}$

 

$\color{red}{Q) \mathbf{About\; Suffix\; Lengths : }}$

$\color{green}{i) \text{ Length of the suffix  = n  :  }}\color{blue}{\text{  n+1}}$

$\color{green}{ii) \text{ Length of the suffix  = m & total length of the string exactly = n :  }}\color{blue}{\text{ n+2}}\color{magenta}{\text{ [ note:- n ≥ m ]}}$

$\color{green}{iii) \text{ Length of the suffix  = m & total length of the string atmost = n :  }}\color{blue}{\text{ (n-m+1)*(m+1)+1}}\color{magenta}{\text{ [ note:- n ≥ m ]}}$

$\color{green}{iv) \text{ Length of the suffix  = m & total length of the string atleast = n :  }}\color{blue}{\text{  max(m,n)+1}}$

$\color{green}{v) \text{ Length of the suffix  = m & total length of the string divisible by n :  }}\color{blue}{\text{ m+n}}$

 

$\color{red}{Q) \mathbf{About\; Substring\; Lengths : }}$

$\color{green}{i) \text{ Length of the substring  = n  :  }}\color{blue}{\text{  n+1}}$

$\color{green}{ii) \text{ Length of the substring  = m & total length of the string exactly = n :  }}\color{blue}{\text{ (n-m+1)*(m+1)+1}}\color{magenta}{\text{ [ note:- n ≥ m ]}}$

$\color{green}{iii) \text{ Length of the substring  = m & total length of the string atmost = n :  }}\color{blue}{\text{ (n-m+1)*(m+1)+1}}\color{magenta}{\text{ [ note:- n ≥ m ]}}$

$\color{green}{iv) \text{ Length of the substring  = m & total length of the string atleast = n :  }}\color{blue}{\text{  (n-m+1)*(m+1)}}$

$\color{green}{v) \text{ Length of the substring  = m & total length of the string divisible by n :  }}\color{blue}{\text{ (n)*(m+1)}}$

 

$\color{red}{Q) \mathbf{About\; Fixed\; position : }}$

$\color{green}{i) \;\;n^{th}\text{ position from the left side of the string is fixed to ‘0’  : }}\color{blue}{\text{  n+2}}$

$\color{green}{ii) \;\;n^{th}\text{ position from the right side of the string is fixed to ‘0’  : }}\color{blue}{\text{ }2^{n}}$

 

$\color{red}{Q) \mathbf{About\; Number \;of\;occurances : }}$

$\color{green}{i) \text{ Number of ZERO’s the strings exactly = n  :  }}\color{blue}{\text{  n+2}}$

$\color{green}{ii) \text{ Number of ZERO’s in the strings atmost = n  :  }}\color{blue}{\text{  n+2}}$

$\color{green}{iii) \text{ Number of ZERO’s in the strings atleast = n  :  }}\color{blue}{\text{  n+1}}$

$\color{green}{iv) \text{ Number of ZERO’s in the strings divisible by n  :  }}\color{blue}{\text{  n}}$

 

let two positive integers m and n, neither m divides n nor n divides m:

$\color{green}{v) \text{ Number of ZERO’s in the strings divisible by n and m  :  }}\color{blue}{\text{ LCM(n,m) }}$

$\color{green}{vi) \text{ Number of ZERO’s in the strings divisible by n or m  :  }}\color{blue}{\text{ LCM(n,m) }}$

$\color{green}{vii) \text{ Number of ZERO’s in the strings divisible by n but not by m  :  }}\color{blue}{\text{ LCM(n,m) }}$

 

let two positive integers m and n, either m divides n or n divides m:

$\color{green}{viii) \text{ Number of ZERO’s in the strings divisible by n and m  :  }}\color{blue}{\text{ MAX(n,m) }}$

$\color{green}{ix) \text{ Number of ZERO’s in the strings divisible by n or m  :  }}\color{blue}{\text{ MIN(n,m) }}$

$\color{green}{v) \text{ Number of ZERO’s in the strings divisible by n but not by m  :  }}\color{blue}{\text{ MAX(n,m) }}$

 

$\color{red}{Q) \mathbf{About\; independent Data items : }}$

While multiplying, you should take care of productive states.

$\color{green}{i) \text{ no.of ZERO’s are exactly ‘n’ and no.of ONE’s exactly ‘m’  : }}$

$\;\;\;\color{blue}{ \text{ Number of ZERO’s the strings exactly = n  :  }}\color{blue}{\text{  n+2}} = \color{magenta}{\text{  (n+1)+(1 DS)}}$

$\;\;\;\color{blue}{ \text{ Number of ONE’s the strings exactly = m  :  }}\color{blue}{\text{  m+2}} = \color{magenta}{\text{  (m+1)+(1 DS)}}$

$\color{DarkOrange}{\text{Answer = (n+1)*(m+1) + 1}}$

 

$\color{green}{ii) \text{ no.of ZERO’s are exactly ‘n’ and no.of ONE’s atleast ‘m’  : }}$

$\;\;\;\color{blue}{ \text{ Number of ZERO’s the strings exactly = n  :  }}\color{blue}{\text{  n+2}} = \color{magenta}{\text{  (n+1)+(1 DS)}}$

$\;\;\;\color{blue}{ \text{ Number of ONE’s the strings exactly = m  :  }}\color{blue}{\text{  m+2}} = \color{magenta}{\text{  (m+1)}}$

$\color{DarkOrange}{\text{Answer = (n+1)*(m+1) + 1}}$

 

Supported Documents : https://drive.google.com/open?id=1haPRlS78p7RA7ugMhaK9Li6370ZjaEcx

posted 4 days ago in Theory of Computation by Veteran (63,971 points) | 238 views
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Great effort!!

Thank you so much
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