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.
$\color{green}{i) \text{ no.of ZERO’s are exactly ‘n’ and no.of ONE’s exactly ‘m’ : }}$
$\;\;\;\color{blue}{ \text{ Number of ZERO’s the strings exactly = n : }}\color{blue}{\text{ n+2}} = \color{magenta}{\text{ (n+1)+(1 DS)}}$
$\;\;\;\color{blue}{ \text{ Number of ONE’s the strings exactly = m : }}\color{blue}{\text{ m+2}} = \color{magenta}{\text{ (m+1)+(1 DS)}}$
$\color{DarkOrange}{\text{Answer = (n+1)*(m+1) + 1}}$
$\color{green}{ii) \text{ no.of ZERO’s are exactly ‘n’ and no.of ONE’s atleast ‘m’ : }}$
$\;\;\;\color{blue}{ \text{ Number of ZERO’s the strings exactly = n : }}\color{blue}{\text{ n+2}} = \color{magenta}{\text{ (n+1)+(1 DS)}}$
$\;\;\;\color{blue}{ \text{ Number of ONE’s the strings exactly = m : }}\color{blue}{\text{ m+2}} = \color{magenta}{\text{ (m+1)}}$
$\color{DarkOrange}{\text{Answer = (n+1)*(m+1) + 1}}$
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