Can't Understand this line in Pascal's Triangle for C++

Question

I have to write a code that generates a pascal's triangle with 12 rows.

I've written everything on my own except one part, That's the formula we use to generate numbers. And the problem is I don't understand what's the connection between our counters and generated numbers (since we're using our counters.).

#include <iostream>
#include <string>

using namespace std;

int main() {
    const int rows=12;
    int padding, value, fxValue;

    for(int rowCounter=0; rowCounter<rows; rowCounter++)
    {
        fxValue=1;
        cout << string((rows-rowCounter)*6, ' ');

        for(int fxCounter=0; fxCounter<=rowCounter; fxCounter++)
        {
            value=fxValue;
            fxValue = fxValue*(rowCounter-fxCounter)/(fxCounter+1);

//          cout << "fxCounter: "<< fxCounter << endl
//               << "rowCounter: " << rowCounter << endl
//               << "fxCounter: " << fxCounter << endl
//               << "fxValue: " << fxValue << endl;

            padding=fxValue/10;

            if(padding==0) cout << value << string(11, ' ');
            else if(10>padding) cout << value << string(10, ' ');
            else if(padding>10) cout << value << string(9, ' ');
        }
        cout << endl;
    }
    return 0;
}

Here's the problem:

fxValue = fxValue*(rowCounter-fxCounter)/(fxCounter+1);

Can someone please explain how did the author came up with the idea of using these variables and how it works fine?

Answer 1

This works because Pascal's triangle can be represented with binomial coefficients :

This formula in your code is based on the fact that, on the same n-index (in the pascal's triangle case, same line), in order to get the next element (k -> k+1), we need to multiply the current value by (n-k)/(k+1):

It's fairly easy to prove if you want to convince yourself about that. So you can get the next value from the previous one with this operation.