My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

       
#include <vector>
       
#include <iostream>
       
 
       
int main()
       
{
       
  // maximum index n and/or k
       
  unsigned int       maxN      = 100;
       
  // what is considered "big" ?
       
  unsigned long long maxNumber = 1000000;
       
  std::cin >> maxN >> maxNumber;
       
 
       
  // this will be the displayed result
       
  unsigned int bigNumbers = 0;
       
 
       
  // setup a 2D array to hold all values
       
  std::vector<std::vector<unsigned long long>> combinations(maxN + 1);
       
 
       
  // C(n,0) = C(n,n) = 1
       
  for (unsigned int n = 0; n <= maxN; n++)
       
  {
       
    combinations[n].resize(n + 1, 0);
       
    combinations[n][0] = combinations[n][n] = 1;
       
  }
       
 
       
  // recursive definition:
       
  // C(n,k) = C(n-1, k-1) + C(n-1, k)
       
  for (unsigned int n = 1; n <= maxN; n++)
       
    for (unsigned int k = 1; k < n; k++)
       
    {
       
      auto sum = combinations[n - 1][k - 1] + combinations[n - 1][k];
       
      // clamp numbers to avoid exceeding 64 bits
       
      if (sum > maxNumber)
       
      {
       
        sum = maxNumber + 1;
       
        // we found one more big number
       
        bigNumbers++;
       
      }
       
 
       
      // store result
       
      combinations[n][k] = sum;
       
    }
       
 
       
  std::cout << bigNumbers << std::endl;
       
  return 0;
       
}

This solution contains 7 empty lines, 10 comments and 2 preprocessor commands.