# Find numbers whose sum of digits equals a value

Data StructureDynamic ProgrammingAlgorithms

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There are a number n and a value. We have to find all n digit numbers, where the sum of all n digit the s is the same as the given value. Here 0 is noa t counted as a digit.

The number n must be in the range 1 to 100, and value must be in range 1 to 500.

## Input and Output

Input:
This algorithm takes number of digits, and the sum value.
Let the digit count is 3. and sum is 15.
Output:
Display the number of different 3-digit numbers whose sum is 15.
The result is 69. (There are 69 different 3-digit numbers whose sum is 15)

## Algorithm

count(digit, sum)

Input: Digit of numbers, given value.

Output: Count how many numbers.

Begin
if digit = 0, then
return true when sum = 0

if memTable[digit, sum] is not vacant, then
return memTable[digit, sum]

for i := 0 to 9 do
if sum – i >= 0, then
answer := answer + count(digit – 1, sum - i)
done

return memTable[digit, sum] := answer
End

numberCount(digit, sum)

Input: Digits of number, the given value.

Output: How many numbers of that type of number.

Begin
define memTable and make all space vacant
res := 0

for i := 1 to 9, do
if sum – i >= 0, then
res := res + count(digit – 1, sum - i)
done

return result
End

## Example

#include<iostream>
#define ROW 101
#define COL 501
using namespace std;

unsigned long long int memTable[ROW][COL];

unsigned long long int count(int digit, int sum) {
if (digit == 0)    //for 0 digit number check if the sum is 0 or not
return sum == 0;

if (memTable[digit][sum] != -1)    //when subproblem has found the result, return it
return memTable[digit][sum];

unsigned long long int ans = 0;    //set the answer to 0 for first time

for (int i=0; i<10; i++)    //count for each digit and find numbers starting with it
if (sum-i >= 0)
ans += count(digit-1, sum-i);
return memTable[digit][sum] = ans;
}

unsigned long long int numberCount(int digit, int sum) {
for(int i = 0; i<ROW; i++)    //initially all entries of memorization table is -1
for(int j = 0; j<ROW; j++)
memTable[i][j] = -1;

unsigned long long int result = 0;
for (int i = 1; i <= 9; i++)    //count for each digit and find numbers starting with it
if (sum-i >= 0)
result += count(digit-1, sum-i);
return result;
}

int main() {
int digit, sum;
cout << "Enter digit count: "; cin >> digit;
cout << "Enter Sum: "; cin >> sum;
cout << "Number of values: " << numberCount(digit, sum);
}

## Output

Enter digit count: 3
Enter Sum: 15
Number of values: 69