Count of n digit numbers whose sum of digits equals to given sum - GeeksforGeeks


Count of n digit numbers whose sum of digits equals to given sum - GeeksforGeeks
Given two integers 'n' and 'sum', find count of all n digit numbers with sum of digits as 'sum'. Leading 0's are not counted as digits.
1 <= n <= 100 and 1 <= sum <= 50000

   countRec(n, sum) = ∑countRec(n-1, sum-x)
                        where 0 =< x <= 9 and
                        sum-x >= 0
   One important observation is, leading 0's must be
   handled explicitly as they are not counted as digits.
   So our final count can be written as below.
  
   finalCount(n, sum) = ∑countRec(n-1, sum-x)
                         where 1 =< x <= 9and
                               sum-x >= 0
// Recursive function to count 'n' digit numbers
// with sum of digits as 'sum'. This function
// considers leading 0's also as digits, that is
// why not directly called
unsigned long long int countRec(int n, int sum)
{
    // Base case
    if (n == 0)
       return sum == 0;
 
    // Initialize answer
    unsigned long long int ans = 0;
 
    // Traverse through every digit and count
    // numbers beginning with it using recursion
    for (int i=0; i<=9; i++)
       if (sum-i >= 0)
          ans += countRec(n-1, sum-i);
 
    return ans;
}
 
// This is mainly a wrapper over countRec. It
// explicitly handles leading digit and calls
// countRec() for remaining digits.
unsigned long long int finalCount(int n, int sum)
{
    // Initialize final answer
    unsigned long long int ans = 0;
 
    // Traverse through every digit from 1 to
    // 9 and count numbers beginning with it
    for (int i = 1; i <= 9; i++)
      if (sum-i >= 0)
         ans += countRec(n-1, sum-i);
 
    return ans;
}


// Recursive function to count 'n' digit numbers
// with sum of digits as 'sum'. This function
// considers leading 0's also as digits, that is
// why not directly called
unsigned long long int countRec(int n, int sum)
{
    // Base case
    if (n == 0)
       return sum == 0;
 
    // Initialize answer
    unsigned long long int ans = 0;
 
    // Traverse through every digit and count
    // numbers beginning with it using recursion
    for (int i=0; i<=9; i++)
       if (sum-i >= 0)
          ans += countRec(n-1, sum-i);
 
    return ans;
}
 
// This is mainly a wrapper over countRec. It
// explicitly handles leading digit and calls
// countRec() for remaining digits.
unsigned long long int finalCount(int n, int sum)
{
    // Initialize final answer
    unsigned long long int ans = 0;
 
    // Traverse through every digit from 1 to
    // 9 and count numbers beginning with it
    for (int i = 1; i <= 9; i++)
      if (sum-i >= 0)
         ans += countRec(n-1, sum-i);
 
    return ans;
}
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