Count numbers with exactly K non-zero digits and distinct odd digit sum

19 June 2025

0

Given an Integer N, and a number K, the task is to find out the total numbers from 0 to N which have exactly K non zero digits and the sum of those digits should be odd and that sum should be distinct. The number N can be as large as 10^18.
Examples:

Input : N = 10, K = 1
Output : 5
The numbers which follow the conditions are ->
1, 3, 5, 7 and 9
The digit sum of 10 that is (1+0) = 1 is also odd, but 1 is already included in our count.
Input : N = 100, K = 2
Output : 8

Prerequisites: Digit-DP
Naive Approach:
A naive approach for linear traversing in O(N) of all elements from 0 to N and calculating sum of digits in log(n) where n is the number of digits in that number will fail for large inputs of N.
Efficient Approach:

We can use Dynamic Programming and it’s very useful technique that is digit-dp to solve this problem.
So instead of keeping a record of non-zero integers, we keep a record of zeroes we can keep at different indices, idx in variable K. The number of zeroes we can keep can be found initially by subtracting K with the number of digits in N.
We keep all the digits of N into a vector say, digits.
Now, we calculate range of elements we can keep at index idx by analysing K.
- Suppose at index idx, we are left with K = 1 (A Non-zero value), then our range to put elements is [0, j] where j is the upper bound decided by the tight value obtained from the current index of the digit from vector digits.
- If at idx, we are left with K = 0, then our range becomes [1, j] because we can’t put in 0 there.
Now, also take a parameter that is sum, which will calculate the sum of digits of a number till the base case hits successfully.
Also, a boolean map is used which will store all the odd sums calculated already, so it gives distinct odd sums.
The recurrence will be:
$f(idx, K, tight, sum) = $\sum_{i=0}^{j} f(idx+1, K-1, newtight, sum+i) f(idx, K, tight, sum) = $\sum_{i=1}^{j} f(idx+1, K, newtight, sum+i)$
where j = digits[idx] if tight = 0, else j = 9
Base Case:
- When idx = digits.size(), K == 0 and sum is odd.
  We mark the sum as true and return 1 else return 0.
- If idx > digits.size() then return 0.

So we create a DP table say DP[idx][K][tight][sum] which will store our result from the recurrence above and return the count by memoizing it to this DP table.
Below is the implementation of the above approach:

C++

// C++ program to Count the numbers having
// exactly K non-zero digits and sum
// of digits are odd and distinct.
#include <bits/stdc++.h>
using namespace std;
 
// To store digits of N
vector<int> digits;
 
// visited map
bool vis[170] = { false };
 
// DP Table
int dp[19][19][2][170];
 
// Push all the digits of N into
// digits vector
void ConvertIntoDigit(int n)
{
    while (n) {
        int dig = n % 10;
        digits.push_back(dig);
        n /= 10;
    }
    reverse(digits.begin(), digits.end());
}
 
// Function returns the count
int solve(int idx, int k,
          int tight, int sum)
{
    // If desired number is formed
    // whose sum is odd
    if (idx == digits.size()
        && k == 0 && sum & 1) {
        // If it is not present in map,
        // mark it as true and return 1
        if (!vis[sum]) {
            vis[sum] = 1;
            return 1;
        }
        // Sum is present in map already
        return 0;
    }
 
    // Desired result not found
    if (idx > digits.size()) {
        return 0;
    }
 
    // If that state is already calculated
    // just return that state value
    if (dp[idx][k][tight][sum]) {
        return dp[idx][k][tight][sum];
    }
 
    // Upper limit
    int j;
    if (tight == 0) {
        j = digits[idx];
    }
    else {
        j = 9;
    }
 
    // To store the count of
    // desired numbers
    int cnt = 0;
 
    // If k is non-zero, i ranges from
    // 0 to j else [1, j]
    for (int i = (k ? 0 : 1);
         i <= j; i++) {
        int newtight = tight;
 
        if (i < j) {
            newtight = 1;
        }
 
        // If current digit is 0, decrement
        // k and recurse sum is not changed
        // as we are just adding 0 that
        // makes no difference
        if (i == 0)
            cnt += solve(idx + 1, k - 1,
                         newtight, sum);
 
        // If i is non zero, then k remains
        // unchanged and value is added to sum
        else
            cnt += solve(idx + 1, k, newtight,
                         sum + i);
    }
 
    // Memoize and return
    return dp[idx][k][tight][sum] = cnt;
}
 
// Driver code
int main()
{
 
    // K is the number of exact non-zero
    // elements to have in number
    int N, k;
    N = 169, k = 2;
 
    // break N into its digits
    ConvertIntoDigit(N);
 
    // We keep record of 0s we need to
    // place in the number
    k = digits.size() - k;
    cout << solve(0, k, 0, 0);
}

Java

// Java program to count the numbers having
// exactly K non-zero digits and sum
// of digits are odd and distinct.
import java.util.*;
 
class GFG{
 
// To store digits of N
static Vector<Integer> digits = new Vector<Integer>();
 
// visited map
static boolean []vis = new boolean[170];
 
// DP Table
static int [][][][]dp = new int[19][19][2][170];
 
// Push all the digits of N into
// digits vector
static void ConvertIntoDigit(int n)
{
    while (n > 0)
    {
        int dig = n % 10;
        digits.add(dig);
        n /= 10;
    }
    Collections.reverse(digits);
}
 
// Function returns the count
static int solve(int idx, int k,
                 int tight, int sum)
{
     
    // If desired number is formed
    // whose sum is odd
    if (idx == digits.size() && 
          k == 0 && sum % 2 == 1) 
    {
         
        // If it is not present in map,
        // mark it as true and return 1
        if (!vis[sum]) 
        {
            vis[sum] = true;
            return 1;
        }
         
        // Sum is present in map already
        return 0;
    }
 
    // Desired result not found
    if (idx > digits.size())
    {
        return 0;
    }
 
    // If that state is already calculated
    // just return that state value
    if (dp[idx][k][tight][sum] > 0) 
    {
        return dp[idx][k][tight][sum];
    }
 
    // Upper limit
    int j;
    if (idx < digits.size() && tight == 0)
    {
        j = digits.get(idx);
    }
    else
    {
        j = 9;
    }
 
    // To store the count of
    // desired numbers
    int cnt = 0;
 
    // If k is non-zero, i ranges from
    // 0 to j else [1, j]
    for(int i = (k > 0 ? 0 : 1); i <= j; i++)
    {
        int newtight = tight;
 
        if (i < j)
        {
            newtight = 1;
        }
 
        // If current digit is 0, decrement
        // k and recurse sum is not changed
        // as we are just adding 0 that
        // makes no difference
        if (i == 0)
            cnt += solve(idx + 1, k - 1,
                         newtight, sum);
 
        // If i is non zero, then k remains
        // unchanged and value is added to sum
        else
            cnt += solve(idx + 1, k, newtight,
                         sum + i);
    }
 
    // Memoize and return
    return dp[idx][k][tight][sum] = cnt;
}
 
// Driver code
public static void main(String[] args)
{
 
    // K is the number of exact non-zero
    // elements to have in number
    int N, k;
    N = 169; k = 2;
 
    // break N into its digits
    ConvertIntoDigit(N);
 
    // We keep record of 0s we need to
    // place in the number
    k = digits.size() - k;
     
    System.out.print(solve(0, k, 0, 0));
}
}
 
// This code is contributed by amal kumar choubey

Python3

# Python3 program to Count the numbers having
# exactly K non-zero digits and sum
# of digits are odd and distinct.
  
# To store digits of N
digits = []
  
# visited map
vis = [False for i in range(170)]
  
# DP Table
dp = [[[[0 for l in range(170)] for k in range(2)] for j in range(19)] for i in range(19)]
  
# Push all the digits of N into
# digits vector
def ConvertIntoDigit(n):
 
    while (n > 0):
        dig = n % 10;
        digits.append(dig);
        n //= 10;
    digits.reverse()
     
# Function returns the count
def solve(idx, k, tight, sum):
 
    # If desired number is formed
    # whose sum is odd
    if (idx == len(digits) and k == 0 and sum % 2 == 1):
         
        # If it is not present in map,
        # mark it as true and return 1
        if (not vis[sum]):
            vis[sum] = True;
            return 1;
         
        # Sum is present in map already
        return 0;
     
    # Desired result not found
    if (idx > len(digits)):
        return 0;
     
    # If that state is already calculated
    # just return that state value
    if (dp[idx][k][tight][sum]):
        return dp[idx][k][tight][sum];
     
    # Upper limit
    j = 0;
    if (idx<len(digits) and tight == 0):
        j = digits[idx];
     
    else:
        j = 9;
     
    # To store the count of
    # desired numbers
    cnt = 0;
  
    # If k is non-zero, i ranges from
    # 0 to j else [1, j]
    for i in range(0 if k else 1, j + 1):
        newtight = tight;
  
        if (i < j):
            newtight = 1;
  
        # If current digit is 0, decrement
        # k and recurse sum is not changed
        # as we are just adding 0 that
        # makes no difference
        if (i == 0):
            cnt += solve(idx + 1, k - 1, newtight, sum);
  
        # If i is non zero, then k remains
        # unchanged and value is added to sum
        else:   
            cnt += solve(idx + 1, k, newtight, sum + i);
     
    dp[idx][k][tight][sum] = cnt
     
    # Memoize and return
    return cnt;
 
# Driver code
if __name__=='__main__':
  
    # K is the number of exact non-zero
    # elements to have in number
    N = 169
    k = 2;
  
    # break N into its digits
    ConvertIntoDigit(N);
  
    # We keep record of 0s we need to
    # place in the number
    k = len(digits) - k;
    print(solve(0, k, 0, 0))
 
# This code is contributed by rutvik_56.

C#

// C# program to count the numbers having
// exactly K non-zero digits and sum
// of digits are odd and distinct.
using System;
using System.Collections.Generic;
 
class GFG{
 
// To store digits of N
static List<int> digits = new List<int>();
 
// visited map
static bool []vis = new bool[170];
 
// DP Table
static int [,,,]dp = new int[ 19, 19, 2, 170 ];
 
// Push all the digits of N into
// digits vector
static void ConvertIntoDigit(int n)
{
    while (n > 0)
    {
        int dig = n % 10;
        digits.Add(dig);
        n /= 10;
    }
    digits.Reverse();
}
 
// Function returns the count
static int solve(int idx, int k,
                 int tight, int sum)
{
     
    // If desired number is formed
    // whose sum is odd
    if (idx == digits.Count && 
          k == 0 && sum % 2 == 1) 
    {
         
        // If it is not present in map,
        // mark it as true and return 1
        if (!vis[sum]) 
        {
            vis[sum] = true;
            return 1;
        }
         
        // Sum is present in map already
        return 0;
    }
 
    // Desired result not found
    if (idx > digits.Count)
    {
        return 0;
    }
 
    // If that state is already calculated
    // just return that state value
    if (dp[idx, k, tight, sum] > 0) 
    {
        return dp[idx, k, tight, sum];
    }
 
    // Upper limit
    int j;
    if (idx < digits.Count && tight == 0)
    {
        j = digits[idx];
    }
    else
    {
        j = 9;
    }
 
    // To store the count of
    // desired numbers
    int cnt = 0;
 
    // If k is non-zero, i ranges from
    // 0 to j else [1, j]
    for(int i = (k > 0 ? 0 : 1); i <= j; i++)
    {
        int newtight = tight;
 
        if (i < j)
        {
            newtight = 1;
        }
 
        // If current digit is 0, decrement
        // k and recurse sum is not changed
        // as we are just adding 0 that
        // makes no difference
        if (i == 0)
            cnt += solve(idx + 1, k - 1,
                         newtight, sum);
 
        // If i is non zero, then k remains
        // unchanged and value is added to sum
        else
            cnt += solve(idx + 1, k, newtight,
                         sum + i);
    }
 
    // Memoize and return
    return dp[idx, k, tight, sum] = cnt;
}
 
// Driver code
public static void Main(String[] args)
{
     
    // K is the number of exact non-zero
    // elements to have in number
    int N, k;
    N = 169; k = 2;
 
    // break N into its digits
    ConvertIntoDigit(N);
 
    // We keep record of 0s we need to
    // place in the number
    k = digits.Count - k;
     
    Console.Write(solve(0, k, 0, 0));
}
}
 
// This code is contributed by amal kumar choubey 

Javascript

<script>
 
    // JavaScript program to count the numbers having
    // exactly K non-zero digits and sum
    // of digits are odd and distinct.
     
    // To store digits of N
    let digits = [];
 
    // visited map
    let vis = new Array(170);
    vis.fill(false);
 
    // DP Table
    let dp = new Array(19);
    for(let i = 0; i < 19; i++)
    {
        dp[i] = new Array(19);
        for(let j = 0; j < 19; j++)
        {
            dp[i][j] = new Array(2);
            for(let k = 0; k < 2; k++)
            {
                dp[i][j][k] = new Array(170);
                for(let l = 0; l < 170; l++)
                {
                    dp[i][j][k][l] = 0;
                }
            }
        }
    }
 
    // Push all the digits of N into
    // digits vector
    function ConvertIntoDigit(n)
    {
        while (n > 0)
        {
            let dig = n % 10;
            digits.push(dig);
            n = parseInt(n / 10, 10);
        }
        digits.reverse();
    }
 
    // Function returns the count
    function solve(idx, k, tight, sum)
    {
 
        // If desired number is formed
        // whose sum is odd
        if (idx == digits.length &&
              k == 0 && sum % 2 == 1)
        {
 
            // If it is not present in map,
            // mark it as true and return 1
            if (!vis[sum])
            {
                vis[sum] = true;
                return 1;
            }
 
            // Sum is present in map already
            return 0;
        }
 
        // Desired result not found
        if (idx > digits.length)
        {
            return 0;
        }
 
        // If that state is already calculated
        // just return that state value
        if (dp[idx][k][tight][sum] > 0)
        {
            return dp[idx][k][tight][sum];
        }
 
        // Upper limit
        let j;
        if (idx < digits.length && tight == 0)
        {
            j = digits[idx];
        }
        else
        {
            j = 9;
        }
 
        // To store the count of
        // desired numbers
        let cnt = 0;
 
        // If k is non-zero, i ranges from
        // 0 to j else [1, j]
        for(let i = (k > 0 ? 0 : 1); i <= j; i++)
        {
            let newtight = tight;
 
            if (i < j)
            {
                newtight = 1;
            }
 
            // If current digit is 0, decrement
            // k and recurse sum is not changed
            // as we are just adding 0 that
            // makes no difference
            if (i == 0)
                cnt += solve(idx + 1, k - 1,
                             newtight, sum);
 
            // If i is non zero, then k remains
            // unchanged and value is added to sum
            else
                cnt += solve(idx + 1, k, newtight,
                             sum + i);
        }
 
        // Memoize and return
        dp[idx][k][tight][sum] = cnt;
        return dp[idx][k][tight][sum];
    }
     
    // K is the number of exact non-zero
    // elements to have in number
    let N, k;
    N = 169; k = 2;
  
    // break N into its digits
    ConvertIntoDigit(N);
  
    // We keep record of 0s we need to
    // place in the number
    k = digits.length - k;
      
    document.write(solve(0, k, 0, 0));
 
</script>

Output:

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Count numbers with exactly K non-zero digits and distinct odd digit sum

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