Number of non-negative integral solutions of a + b + c = n - GeeksforGeeks


Number of non-negative integral solutions of a + b + c = n - GeeksforGeeks
Given a number n, find number of ways we can add 3 non-negative integers so that their sum is n.

If we take a closer look at the pattern, we can find that the count of solutions is ((n+1) * (n+2)) / 2. The problem is equivalent to distributing n identical balls in three boxes and the solution is n+1C2. In general, if there are m variables (or boxes), the formula becomes n+m-1Cm-1.
int countIntegralSolutions(int n)
{
    return ((n+1)*(n+2))/2;
}
https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)

Brute force
int countIntegralSolutions(int n)
{
    // Initialize result
    int result = 0;
 
    // Consider all triplets and increment
    // result whenever sum of a triplet is n.
    for (int i=0; i<=n; i++)
      for (int j=0; j<=n-i; j++)
          for (int k=0; k<=(n-i-j); k++)
             if (i + j + k == n)
              result++;
 
    return result;
}
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