anyone please? 148 ans is given but i am getting 150.. in given solution the total no of string possible is n…

Comments

• KG AI MentorKG AI Mentor9 Feb 2026

  Let's carefully analyze the problem and resolve the confusion. --- Problem Statement Recap:We have a string $ w $ of length 17, with positions numbered 1 to 17. Constraints: Characters at positions 2 and 10 are the same. Characters at positions 5, 8, and 11 are the same. All other positions (i.e., positions 1, 3, 4, 6, 7, 9, 12, 13, 14, 15, 16, 17) contain unique characters, different from each other and from the characters at positions 2/10 and 5/8/11. We are to find: How many distinct substrings are possible in such a string? --- Step 1: Understand the structure of the stringLet’s denote the characters: Let $ a $ be the character at positions 2 and 10. Let $ b $ be the character at positions 5, 8, and 11. Let $ c1, c2, \dots, c{12} $ be the characters at the remaining 12 positions (1, 3, 4, 6, 7, 9, 12, 13, 14, 15, 16, 17), all distinct and different from $ a $ and $ b $. So, the string has: 17 positions. 2 positions with $ a $: pos 2 and 10. 3 positions with $ b $: pos 5, 8, 11. 12 positions with unique characters (all different from each other and from $ a $, $ b $). So, total distinct characters in the string: $ 1 (a) + 1 (b) + 12 (unique) = 14 $ distinct characters. But the key is: We are not counting the number of such strings, but rather: For a string satisfying these constraints, how many distinct substrings can exist? Wait — actually, the question is: How many distinct substrings are possible in such a string? But this is ambiguous: are we to count the number of distinct substrings that can appear in any such string (i.e., over all possible strings satisfying the constraints), or are we to assume a fixed string satisfying the constraints and count the number of distinct substrings in it? But the answer is given as 148, and the user says they get 150, so likely the question is: For a fixed string satisfying the constraints, how many distinct substrings are there? But that can't be — because the number of distinct substrings depends on the actual characters. For example, if the string is "a b c d e f g h i j k l m n o p q", with constraints, but the characters are chosen so that no substring repeats, then the number of distinct substrings is maximized. But the constraints force some characters to be equal, so some substrings will be the same. But the question is: How many distinct substrings are possible in such a string? Wait — the phrasing is ambiguous. But given that the answer is a fixed number (148), and the user is getting 150, it suggests that the question is: What is the maximum number of distinct substrings possible in such a string? Or perhaps: What is the number of distinct substrings in a string satisfying the constraints, assuming all other characters are distinct and different from a and b? But actually, the key is: The string has constraints on equality of characters, but all other characters are unique. So, the only repeated characters are: $ w[2] = w[10] = a $ $ w[5] = w[8] = w[11] = b $ All other positions have unique characters. So, the string has only two repeated characters: $ a $ appears twice, $ b $ appears three times. Now, the number of distinct substrings in a string of length $ n $ is at most $ \frac{n(n+1)}{2} $, which is $ \frac{17 \cdot 18}{2} = 153 $. But because of repeated characters, some substrings will be the same, so the number of distinct substrings will be less than 153. But the user says they get 150, and the answer is 148. So likely, the question is: What is the number of distinct substrings in such a string, assuming that all characters not forced to be equal are distinct and different from each other and from $ a $ and $ b $? And we are to compute the maximum possible number of distinct substrings under these constraints. But actually, the number of distinct substrings depends on the arrangement of the characters. But the constraints only fix which positions have equal characters, not their values. But to maximize the number of distinct substrings, we should assume that the characters are chosen so that no two substrings are equal unless forced by the constraints. So, we can assume that: $ a $ and $ b $ are distinct. All other characters are distinct and different from $ a $, $ b $. The only equal substrings are those that are forced by the equal characters. So, we need to compute the number of distinct substrings in such a string, assuming that all substrings that are not forced to be equal are distinct. So, the total number of substrings is $ \frac{17 \cdot 18}{2} = 153 $. We subtract the number of duplicate substrings — i.e., substrings that are equal due to the constraints. So, we need to find how many pairs of substrings are equal because of the repeated characters. But actually, we need to find how many distinct substrings are there, so we need to subtract the number of redundant substrings — i.e., substrings that are equal to others. But it's easier to compute the total number of substrings, and subtract the number of duplicate substrings — but we have to be careful not to double-count. Alternatively, we can compute the number of distinct substrings by considering that the only equal substrings are those that are forced by the equal characters. So, let's proceed. --- Step 2: Total number of substringsTotal number of substrings in a string of length 17: $$ \frac{17 \cdot 18}{2} = 153 $$ Now, we need to subtract the number of duplicate substrings — i.e., substrings that are equal to others due to the constraints. But actually, we need to find how many distinct substrings there are. So, we need to find how many pairs of substrings are equal, and then subtract the overcount. But a better way is to consider that if two substrings are equal, they must be equal because of the repeated characters. So, we need to find all pairs of substrings that are forced to be equal due to the constraints. Let’s consider the repeated characters: $ w[2] = w[10] = a $ $ w[5] = w[8] = w[11] = b $ All other characters are unique. So, the only possible equal substrings are those that: Start or end at positions where characters are equal, and The substrings are identical due to the equal characters. But since all other characters are unique, any substring that contains a character from a unique position will be unique, unless it is forced to be equal by the repeated characters. So, the only substrings that can be equal are those that: Are identical in content due to the repeated characters. Let’s consider the only possible equal substrings: --- Case 1: Substrings that are equal because $ w[2] = w[10] = a $We need to find substrings that are equal because they both contain $ a $ at positions 2 and 10, but otherwise are the same. But for two substrings to be equal, they must have the same characters in the same order. So, suppose we have a substring $ s $ that starts at position $ i $ and ends at $ j $, and another substring $ s' $ that starts at $ i' $ and ends at $ j' $, and $ s = s' $. But since all characters except $ a $ and $ b $ are unique, the only way two substrings can be equal is if they are identical in content, which can only happen if the characters at corresponding positions are equal. So, for example, if a substring includes $ w[2] $, and another substring includes $ w[10] $, and the rest of the characters are the same, then the substrings could be equal. But since $ w[2] = w[10] $, and all other characters are unique, the only way two substrings can be equal is if they are identical in content, and the only possible equal characters are $ a $ and $ b $. So, we need to find all pairs of substrings that are equal due to the repeated characters. But actually, we can think: How many substrings are duplicated? But it's better to compute the number of distinct substrings by considering that the only equal substrings are those that are forced by the repeated characters. Let’s consider the only possible equal substrings: --- Substrings that are equal due to $ w[2] = w[10] = a $Suppose we have a substring that includes $ w[2] $, and another substring that includes $ w[10] $, and the rest of the characters are the same. But for the substrings to be equal, the characters at corresponding positions must be equal. So, for example, consider a substring that starts at position $ i $ and ends at $ j $, and another substring that starts at $ i' $ and ends at $ j' $, such that $ w[i] = w[i'] $, $ w[i+1] = w[i'+1] $, etc. But since all characters except $ a $ and $ b $ are unique, the only way this can happen is if the substrings are shifted by the difference between positions 2 and 10. Let’s define the shift between position 2 and 10: $ 10 - 2 = 8 $. So, if we have a substring that starts at position $ i $ and ends at $ j $, and another substring that starts at $ i+8 $ and ends at $ j+8 $, and the characters at $ i $ and $ i+8 $ are equal, etc., then the substrings could be equal. But in our case, the only equal characters are $ w[2] = w[10] $, and $ w[5] = w[8] = w[11] $. So, the only possible equal substrings are those that are shifted by 8 (from 2 to 10), or by 3 (from 5 to 8), or by 6 (from 5 to 11), etc. But let’s consider the only possible equal substrings: --- Substrings that are equal due to $ w[2] = w[10] $Let’s consider substrings that start at position $ i $ and end at $ j $, and another substring that starts at $ i+8 $ and ends at $ j+8 $, and $ w[i] = w[i+8] $, etc. But the only positions where characters are equal are: $ w[2] = w[10] $ $ w[5] = w[8] = w[11] $ So, for a substring to be equal to another substring shifted by 8, we need that for all $ k $, $ w[i+k] = w[i+k+8] $. But this can only happen if the substring is entirely within the range where the characters are equal. But since only $ w[2] = w[10] $, and all other characters are unique, the only way $ w[i+k] = w[i+k+8] $ is if $ i+k = 2 $ and $ i+k+8 = 10 $, or vice versa. So, the only possible equal substrings due to $ w[2] = w[10] $ are those that are identical and involve the same characters. But actually, the only way two substrings can be equal is if they are identical in content, and the only possible equal characters are $ a $ and $ b $. So, let’s consider the only possible equal substrings: --- Substrings that are equal due to $ w[2] = w[10] $Suppose we have a substring $ s $ that starts at position $ i $ and ends at $ j $, and another substring $ s' $ that starts at $ i+8 $ and ends at $ j+8 $, and $ s = s' $. For this to happen, we need $ w[i+k] = w[i+k+8] $ for all $ k $ from 0 to $ j-i $. But since only $ w[2] = w[10] $, and all other characters are unique, the only way this can happen is if the substring is of length 1, and $ i = 2 $, $ i+8 = 10 $, so $ w[2] = w[10] $, so the substrings of length 1 at positions 2 and 10 are equal. But for longer substrings, we need more equal characters. For example, a substring of length 2: $ w[2]w[3] $ and $ w[10]w[11] $. But $ w[3] $ is unique, $ w[11] = b $, so $ w[3] \neq w[11] $, so $ w[2]w[3] \neq w[10]w[11] $. Similarly, $ w[1]w[2] $ and $ w[9]w[10] $: $ w[1] $ is unique, $ w[9] $ is unique, and $ w[1] \neq w[9] $, so $ w[1]w[2] \neq w[9]w[10] $. So, the only possible equal substrings due to $ w[2] = w[10] $ are the single-character substrings at positions 2 and 10. Similarly, for $ w[5] = w[8] = w[11] $, the only possible equal substrings are the single-character substrings at positions 5, 8, 11. But wait — could there be longer substrings that are equal? For example, consider a substring that includes $ w[5] $ and $ w[8] $, but since $ w[5] = w[8] $, and the characters in between are unique, it's unlikely. But let’s check. Suppose we have a substring $ s $ that starts at position $ i $ and ends at $ j $, and another substring $ s' $ that starts at $ i+3 $ and ends at $ j+3 $, and $ s = s' $. For this to happen, we need $ w[i+k] = w[i+k+3] $ for all $ k $. But the only positions where characters are equal are: $ w[5] = w[8] = w[11] $ $ w[2] = w[10] $ So, for $ w[i+k] = w[i+k+3] $, we need $ i+k = 5 $, $ i+k+3 = 8 $, so $ i = 2 $, $ k = 3 $, so $ w[5] = w[8] $. Similarly, $ i+k = 8 $, $ i+k+3 = 11 $, so $ i = 5 $, $ k = 3 $, so $ w[8] = w[11] $. So, for a substring of length 1: $ w[5] = w[8] = w[11] $, so the single-character substrings at positions 5, 8, 11 are equal. For a substring of length 2: $ w[5]w[6] $ and $ w[8]w[9] $: $ w[6] $ is unique, $ w[9] $ is unique, and $ w[6] \neq w[9] $, so $ w[5]w[6] \neq w[8]w[9] $. Similarly, $ w[4]w[5] $ and $ w[7]w[8] $: $ w[4] $ is unique, $ w[7] $ is unique, $ w[4] \neq w[7] $, so $ w[4]w[5] \neq w[7]w[8] $. So, the only possible equal substrings are the single-character substrings at positions 2 and 10, and at positions 5, 8, 11. But wait — could there be a substring that is equal to another substring due to the repeated characters? For example, consider the substring $ w[2] $ and $ w[10] $: both are $ a $, so the substrings of length 1 at positions 2 and 10 are equal. Similarly, $ w[5] = w[8] = w[11] = b $, so the substrings of length 1 at positions 5, 8, 11 are equal. But are there any other equal substrings? For example, could a substring of length 2 be equal to another? Suppose we have $ w[2]w[3] $ and $ w[10]w[11] $: $ w[2] = w[10] = a $, $ w[3] $ is unique, $ w[11] = b $, and $ a \neq b $, $ w[3] \neq b $, so $ w[2] — KG AI Mentor 🤖

• Neet Dhameliya31 Dec 2025

  @sambitasingh yes 150 should be correct.