c++ misunderstanding of binomial coeficient recursive function

code:

#include<stdio.h>
int binomialCoeff(int n, int k)
{
  // Base Cases
  if (k==0 || k==n)
    return 1;
 else
  return  binomialCoeff(n-1, k-1) + binomialCoeff(n-1, k);
}
int main()
{
    int n = 5, k = 2;
    printf("Value of C(%d, %d) is %d ", n, k, binomialCoeff(n, k));
    return 0;
}

I think I can understand the base case. When we use 0 for n and k=n the result is 0!/0! which is = 1. So we return 1. Formula

But I can't understand this part of the code:

 return  binomialCoeff(n-1, k-1) + binomialCoeff(n-1, k);

With values of 5 for n and 2 for k I get the result 10. (When replace in the formula).formula But why we use аddition?

And one more thing. Why the program doesn't work when I set "n" and "k" from the keyboard? Like this:

int main()
{
    int n,k;
    cin>>n;
    cin>>k;
    printf("Value of C(%d, %d) is %d ", n, k, binomialCoeff(n, k));
    return 0;
}

One of the properties of binomial coefficients is that: C(n, k) can be written as C(n-1,k-1) + C(n-1,k) for all 1 <= k <= n-1.

So a C(3,2) = C(2,1) + C(2,2) Or 3 = 2 + 1

This is what is used in the simple recursion example you have shared.

Regarding your second problem, not sure why you should have any problems.