/*binomial.cpp
Thsi function takes an x, y, and positive integer n and calculates (x+y)^n using the binomial theorem.
*/

#include<iostream>
#include<cmath>
using namespace std;

  int findfactorial(double num)
  {

    double fact(1);
    int i(1);
 
    while(i<=num)
    {
      fact*=i;
      i+=1;

    }
    return fact;
  }

double nk(double n, double k)
{

  double numerator, denominator, final;

  numerator = findfactorial(n);
  denominator = findfactorial(k)*findfactorial(n-k);

  final = numerator/denominator;
  return final;

}
int main()
{
  double x, y, sum(0), n;
  cout<<"This function takes in a numerical value for x, a numerical value for y,\nand a positive integer for the exponent n to compute (x+y)^n using the binomial theorem."<<endl<<endl;

  cout<<"Please input your values."<<endl;
  cout<<"x= ";
  cin>>x;
  cout<<"y= ";
  cin>>y;
  cout<<"n= ";
  cin>>n;

  if (x==0&y==0&n==0)
    {
      cout<<"This is indeterminate, thanks for playing!"<<endl;
      return 0;
    }

  if (n<0)
    {
      cout<<"This function only works for exponents that are greater than zero.  Please try again!"<<endl;
      return 0;
    }

  if (ceil(n)!=n)
    {
      cout<<"This function only works for integer values of i.  Please try again!"<<endl;
      return 0;
    }

  for (int k = 0; k<=n; k+=1)
    {
      sum = sum+nk(n, k)*pow(x,n-k)*pow(y, k);



    }
  cout<<"Using the binomial theorem, we find that ("<<x<<" + "<<y<<")^"<<n<<"= "<<sum<<"."<<endl;

  cout<<"We can confirm this by using the built-in power function and we get: "<<pow(x+y, n)<<"."<<endl;

  return 0;
}