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# Count of numbers from range [L, R] whose sum of digits is Y | Set 2

Given three positive integers L, R and Y, the task is to count the numbers in the range [L, R] whose sum of digits is equal to Y

Examples:

Input: L = 500, R = 1000, Y = 6
Output: 3
Explanation:
Numbers in the range [500, 600] whose sum of digits is Y(= 6) are:
501 = 5 + 0 + 1 = 6
510 = 5 + 1 + 0 = 6
600 = 6 + 0 + 0 = 6
Therefore, the required output is 3.

Input: L = 20, R = 10000, Y = 14
Output: 540

Naive Approach: Refer to previous post to solve this problem by iterating over all the numbers in the range [L, R], and for every number, check if its sum of digits is equal to Y or not. If found to be true, then increment the count. Finally, print the count obtained.

Time Complexity: O(R – L + 1) * log10(R)
Auxiliary Space: O(1)

Efficient approach: To optimize the above approach, the idea is to use Digit DP using the following recurrence relation:

where, sum: Represents sum of digits.
tight: Check if sum of digits exceed Y or not.
end: Stores the maximum possible value of ith digit of a number.
cntNum(N, Y, tight): Returns the count of numbers in the range [0, X] whose sum of digits is Y.

Before moving into the DP solution, it is good practice to write down the recursive code.

Here is the recursive code –

## C++

 // C++ Program for the same approach#include using namespace std; // Function to find the sum of digits// of numbers in the range [0, X]int cntNum(string X, int i, int sum, int tight){       // Check if count of digits in a number    // greater than count of digits in X    if (i >= X.length() || sum < 0) {         // Check if sum of digits of a        // number is equal to Y        if (sum == 0) {            return 1;        }         return 0;    }     // Stores count of numbers whose    // sum of digits is Y    int res = 0;     // Check if the number    // exceeds Y or not    int end = tight != 0 ? X[i] - '0' : 9;     // Iterate over all possible    // values of i-th digits    for (int j = 0; j <= end; j++) {         // Update res        res += cntNum(X, i + 1, sum - j,                    (tight > 0 & (j == end)) ==                            true ? 1 : 0);    }     // Return res    return res;}// Utility function to count the numbers in// the range [L, R] whose sum of digits is Ystatic int UtilCntNumRange(int L,int R,int Y){     // Base Case    if (R == 0 && Y == 0) {         return 1;    }// Stores numbers in the form    // of its equivalent String    string str = to_string(R);         // Stores count of numbers    // in the range [0, R]    int cntR = cntNum(str, 0, Y,                    1);     // Update str    str = to_string(L - 1);    // Stores count of numbers in    // the range [0, L - 1]    int cntL = cntNum(str, 0, Y,                    1);     return (cntR - cntL);} // Driver codeint main(){    int L = 20, R = 10000, Y = 14;    cout<<(UtilCntNumRange(L, R, Y));} // This code is contributed by shinjanpatra

## Java

 // Java program for the above approachimport java.util.*;class GFG{// Function to find the sum of digits// of numbers in the range [0, X]static int cntNum(String X, int i, int sum,           int tight) {    // Check if count of digits in a number    // greater than count of digits in X    if (i >= X.length() || sum < 0) {          // Check if sum of digits of a        // number is equal to Y        if (sum == 0) {            return 1;        }          return 0;    }      // Stores count of numbers whose    // sum of digits is Y    int res = 0;      // Check if the number    // exceeds Y or not    int end = tight != 0 ? X.charAt(i) - '0' : 9;      // Iterate over all possible    // values of i-th digits    for (int j = 0; j <= end; j++) {          // Update res        res += cntNum(X, i + 1, sum - j,                      (tight > 0 & (j == end)) ==                               true ? 1 : 0);    }      // Return res    return res; }// Utility function to count the numbers in// the range [L, R] whose sum of digits is Ystatic int UtilCntNumRange(int L,int R,int Y) {       // Base Case    if (R == 0 && Y == 0) {          return 1;    }   // Stores numbers in the form    // of its equivalent String    String str = String.valueOf(R);          // Stores count of numbers    // in the range [0, R]    int cntR = cntNum(str, 0, Y,                      1);      // Update str    str = String.valueOf(L - 1);    // Stores count of numbers in    // the range [0, L - 1]    int cntL = cntNum(str, 0, Y,                      1);      return (cntR - cntL); }// Driver Code public static void main (String[] args)    {      int L = 20, R = 10000, Y = 14;      System.out.print(UtilCntNumRange(L, R, Y));    }}// This code is contributed by Debojyoti Mandal

## Python3

 # Python program for the above approach# Function to find the sum of digits# of numbers in the range [0, X]def cntNum(X, i, sum, tight):     # Check if count of digits in a number    # greater than count of digits in X    if (i >= len(X) or sum < 0):          # Check if sum of digits of a        # number is equal to Y        if (sum == 0):            return 1             return 0      # Stores count of numbers whose    # sum of digits is Y    res = 0      # Check if the number    # exceeds Y or not    end = ord(X[i]) - ord('0') if tight else 9      # Iterate over all possible    # values of i-th digits    for j in range(end+1):          # Update res        res += cntNum(X, i + 1, sum - j,1 if((tight > 0 and (j == end)) == True) else 0)      # Return res    return res # Utility function to count the numbers in# the range [L, R] whose sum of digits is Ydef UtilCntNumRange(L, R, Y):      # Base Case    if (R == 0 and Y == 0):          return 1     # Stores numbers in the form    # of its equivalent String    Str = str(R)      # Stores count of numbers    # in the range [0, R]    cntR = cntNum(Str, 0, Y,1)      # Update str    Str = str(L - 1)         # Stores count of numbers in    # the range [0, L - 1]    cntL = cntNum(Str, 0, Y, 1)      return (cntR - cntL) # Driver Code L, R, Y = 20, 10000, 14print(UtilCntNumRange(L, R, Y)) # This code is contributed by shinjanpatra

## C#

 // C# program for the above approachusing System;class GFG{       // Function to find the sum of digits    // of numbers in the range [0, X]    static int cntNum(string X, int i, int sum, int tight)    {               // Check if count of digits in a number        // greater than count of digits in X        if (i >= X.Length || sum < 0) {             // Check if sum of digits of a            // number is equal to Y            if (sum == 0) {                return 1;            }             return 0;        }         // Stores count of numbers whose        // sum of digits is Y        int res = 0;         // Check if the number        // exceeds Y or not        int end = tight != 0 ? X[i] - '0' : 9;         // Iterate over all possible        // values of i-th digits        for (int j = 0; j <= end; j++) {             // Update res            res += cntNum(                X, i + 1, sum - j,                (tight > 0 & (j == end)) == true ? 1 : 0);        }         // Return res        return res;    }    // Utility function to count the numbers in    // the range [L, R] whose sum of digits is Y    static int UtilCntNumRange(int L, int R, int Y)    {        // Base Case        if (R == 0 && Y == 0) {             return 1;        }        // Stores numbers in the form        // of its equivalent String        string str = R.ToString();         // Stores count of numbers        // in the range [0, R]        int cntR = cntNum(str, 0, Y, 1);         // Update str        str = (L - 1).ToString();        // Stores count of numbers in        // the range [0, L - 1]        int cntL = cntNum(str, 0, Y, 1);         return (cntR - cntL);    }       // Driver Code    public static void Main(string[] args)    {        int L = 20, R = 10000, Y = 14;        Console.WriteLine(UtilCntNumRange(L, R, Y));    }} // This code is contributed by ukasp.

## Javascript

 // JavaScript program for the above approach// Function to find the sum of digits// of numbers in the range [0, X]function cntNum( X, i, sum, tight) {    // Check if count of digits in a number    // greater than count of digits in X    if (i >= X.length || sum < 0) {          // Check if sum of digits of a        // number is equal to Y        if (sum == 0) {            return 1;        }          return 0;    }      // Stores count of numbers whose    // sum of digits is Y    var res = 0;      // Check if the number    // exceeds Y or not    var end = tight != 0 ? X[i].charCodeAt(0) - '0'.charCodeAt(0) : 9;      // Iterate over all possible    // values of i-th digits    for (var j = 0; j <= end; j++) {          // Update res        res += cntNum(X, i + 1, sum - j,                      (tight > 0 & (j == end)) ==                               true ? 1 : 0);    }      // Return res    return res; }// Utility function to count the numbers in// the range [L, R] whose sum of digits is Yfunction UtilCntNumRange(L, R, Y) {     // Base Case    if (R == 0 && Y == 0) {          return 1;    }   // Stores numbers in the form    // of its equivalent String    var str = (R).toString();          // Stores count of numbers    // in the range [0, R]    var cntR = cntNum(str, 0, Y,                      1);      // Update str     str = (L - 1).toString();    // Stores count of numbers in    // the range [0, L - 1]    var cntL = cntNum(str, 0, Y,                      1);      return (cntR - cntL); }// Driver Code       var L = 20, R = 10000, Y = 14;      document.write(UtilCntNumRange(L, R, Y));      // This code is contributed by shivanisinghss2110

Output

540

Follow the steps below to solve the problem using DP.

1. Initialize a 3D array dp[N][Y][tight] to compute and store the values of all subproblems of the above recurrence relation.
2. Finally, return the value of dp[N][sum][tight].

Below is the implementation of the above approach:

## C++

 // CPP program for the above approach#include using namespace std; #define M 1000 // Function to find the sum of digits// of numbers in the range [0, X]int cntNum(string X, int i, int sum,           int tight, int dp[M][M][2]){    // Check if count of digits in a number    // greater than count of digits in X    if (i >= X.length() || sum < 0) {         // If sum of digits of a        // number is equal to Y        if (sum == 0) {            return 1;        }         return 0;    }     // Check if current subproblem has    // already been computed    if (dp[sum][i][tight] != -1) {        return dp[sum][i][tight];    }     // Stores count of numbers whose    // sum of digits is Y    int res = 0;     // Check if the number    // exceeds Y or not    int end = tight ? X[i] - '0' : 9;     // Iterate over all possible    // values of i-th digits    for (int j = 0; j <= end; j++) {         // Update res        res += cntNum(X, i + 1, sum - j,                      (tight & (j == end)), dp);    }     // Return res    return dp[sum][i][tight]=res;} // Utility function to count the numbers in// the range [L, R] whose sum of digits is Yint UtilCntNumRange(int L, int R, int Y){    // Base Case    if (R == 0 && Y == 0) {         return 1;    }     // Stores numbers in the form    // of its equivalent string    string str = to_string(R);     // Stores overlapping subproblems    int dp[M][M][2];     // Initialize dp[][][]    memset(dp, -1, sizeof(dp));     // Stores count of numbers    // in the range [0, R]    int cntR = cntNum(str, 0, Y,                      true, dp);     // Update str    str = to_string(L - 1);     // Initialize dp[][][]    memset(dp, -1, sizeof(dp));     // Stores count of numbers in    // the range [0, L - 1]    int cntL = cntNum(str, 0, Y,                      true, dp);     return (cntR - cntL);} // Driver Codeint main(){    int L = 20, R = 10000, Y = 14;    cout << UtilCntNumRange(L, R, Y);}

## Java

 // Java program for the above approachimport java.util.*; class GFG{ static final int M = 1000; // Function to find the sum of digits// of numbers in the range [0, X]static int cntNum(String X, int i, int sum,           int tight, int dp[][][]){    // Check if count of digits in a number    // greater than count of digits in X    if (i >= X.length() || sum < 0) {         // Check if sum of digits of a        // number is equal to Y        if (sum == 0) {            return 1;        }         return 0;    }     // Check if current subproblem has    // already been computed    if (dp[sum][i][tight] != -1) {        return dp[sum][i][tight];    }     // Stores count of numbers whose    // sum of digits is Y    int res = 0;     // Check if the number    // exceeds Y or not    int end = tight != 0 ? X.charAt(i) - '0' : 9;     // Iterate over all possible    // values of i-th digits    for (int j = 0; j <= end; j++) {         // Update res        res += cntNum(X, i + 1, sum - j,                      (tight > 0 & (j == end)) ==                               true ? 1 : 0, dp);    }     // Return res    return dp[sum][i][tight]=res;} // Utility function to count the numbers in// the range [L, R] whose sum of digits is Ystatic int UtilCntNumRange(int L, int R, int Y){    // Base Case    if (R == 0 && Y == 0) {         return 1;    }     // Stores numbers in the form    // of its equivalent String    String str = String.valueOf(R);     // Stores overlapping subproblems    int [][][]dp = new int[M][M][2];     // Initialize dp[][][]    for(int i = 0; i < M; i++)    {        for (int j = 0; j < M; j++) {            for (int k = 0; k < 2; k++)                dp[i][j][k] = -1;        }    }     // Stores count of numbers    // in the range [0, R]    int cntR = cntNum(str, 0, Y,                      1, dp);     // Update str    str = String.valueOf(L - 1);     // Initialize dp[][][]    for(int i = 0; i < M; i++)    {        for (int j = 0; j < M; j++) {            for (int k = 0; k < 2; k++)                dp[i][j][k] = -1;        }    }     // Stores count of numbers in    // the range [0, L - 1]    int cntL = cntNum(str, 0, Y,                      1, dp);     return (cntR - cntL);} // Driver Codepublic static void main(String[] args){    int L = 20, R = 10000, Y = 14;    System.out.print(UtilCntNumRange(L, R, Y));}} // This code is contributed by shikhasingrajput

## Python3

 # Python program for the above approachM = 1000 # Function to find the sum of digits# of numbers in the range [0, X]def cntNum(X, i, sum, tight, dp):       # Check if count of digits in a number    # greater than count of digits in X    if (i >= len(X) or sum < 0):         # Check if sum of digits of a        # number is equal to Y        if (sum == 0):            return 1         return 0     # Check if current subproblem has    # already been computed    if (dp[sum][i][tight] != -1):        return dp[sum][i][tight]     # Stores count of numbers whose    # sum of digits is Y    res, end = 0, 9     # Check if the number    # exceeds Y or not    if tight:        end = ord(X[i]) - ord('0')    # end = tight ? X[i] - '0' : 9;     # Iterate over all possible    # values of i-th digits    for j in range(end + 1):         # Update res        res += cntNum(X, i + 1, sum - j,                      (tight & (j == end)), dp)     # Return res    dp[sum][i][tight] = res    return res # Utility function to count the numbers in# the range [L, R] whose sum of digits is Ydef UtilCntNumRange(L, R, Y):       # Base Case    if (R == 0 and Y == 0):         return 1     # Stores numbers in the form    # of its equivalent    strr = str(R)     # Stores overlapping subproblems    dp = [[[-1 for i in range(2)] for i in range(M)]                                  for i in range(M)]     # Initialize dp[][][]    # memset(dp, -1, sizeof(dp))     # Stores count of numbers    # in the range [0, R]    cntR = cntNum(strr, 0, Y, True, dp)     # Update str    strr = str(L - 1)     # Initialize dp[][][]    # memset(dp, -1, sizeof(dp))     # Stores count of numbers in    # the range [0, L - 1]    cntL = cntNum(strr, 0, Y, True, dp)     return (cntR - cntL) # Driver Codeif __name__ == '__main__':    L, R, Y = 20, 10000, 14    print(UtilCntNumRange(L, R, Y)) # This code is contributed by mohit kumar 29

## C#

 // C# program for the above approachusing System; class GFG{ static readonly int M = 1000; // Function to find the sum of digits// of numbers in the range [0, X]static int cntNum(String X, int i, int sum,                 int tight, int [,,]dp){         // Check if count of digits in a number    // greater than count of digits in X    if (i >= X.Length || sum < 0)    {                 // Check if sum of digits of a        // number is equal to Y        if (sum == 0)        {            return 1;        }        return 0;    }     // Check if current subproblem has    // already been computed    if (dp[sum, i, tight] != -1)    {        return dp[sum, i, tight];    }     // Stores count of numbers whose    // sum of digits is Y    int res = 0;     // Check if the number    // exceeds Y or not    int end = tight != 0 ? X[i] - '0' : 9;     // Iterate over all possible    // values of i-th digits    for(int j = 0; j <= end; j++)    {                 // Update res        res += cntNum(X, i + 1, sum - j,                    (tight > 0 & (j == end)) ==                      true ? 1 : 0, dp);    }     // Return res    return dp[sum][i][tight] = res;} // Utility function to count the numbers in// the range [L, R] whose sum of digits is Ystatic int UtilCntNumRange(int L, int R, int Y){         // Base Case    if (R == 0 && Y == 0)    {        return 1;    }     // Stores numbers in the form    // of its equivalent String    String str = String.Join("", R);         // Stores overlapping subproblems    int [,,]dp = new int[M, M, 2];     // Initialize [,]dp[]    for(int i = 0; i < M; i++)    {        for(int j = 0; j < M; j++)        {            for(int k = 0; k < 2; k++)                dp[i, j, k] = -1;        }    }     // Stores count of numbers    // in the range [0, R]    int cntR = cntNum(str, 0, Y,                      1, dp);     // Update str    str = String.Join("",L - 1);     // Initialize [,]dp[]    for(int i = 0; i < M; i++)    {        for(int j = 0; j < M; j++)        {            for(int k = 0; k < 2; k++)                dp[i, j, k] = -1;        }    }     // Stores count of numbers in    // the range [0, L - 1]    int cntL = cntNum(str, 0, Y,                      1, dp);     return (cntR - cntL);} // Driver Codepublic static void Main(String[] args){    int L = 20, R = 10000, Y = 14;         Console.Write(UtilCntNumRange(L, R, Y));}} // This code is contributed by 29AjayKumar

## Javascript

 

Output

540

Time Complexity: O(Y * log10(R) * 10)
Auxiliary Space: O(Y * log10(R)