How does this recursive algorithm work?

Question

Java

Hello!
The algorithm for calculating the binomial, as I understand it, is according to the Bernouli formula.

It also uses a recurrent formula for finding the number of combinations

. The algorithm itself:

public double binomial1(int N, int k, double p) {
        if (N == 0 && k == 0) return 1.0;
        if (N < 0 || k < 0) return 0.0;
        return (1.0 - p) *binomial1(N-1, k, p) + p*binomial1(N-1, k-1, p);
    }

Questions:
How is it that we have (1.0 - p) and p in a recurrent expression without powers? After all, according to the Bernoulli formula, there should be. I understand that there is a recursion, apparently, they are multiplied by themselves several times, but I can’t unravel the tangle?
Are there any methods for analyzing such algorithms / formulas, and for creating these formulas. Somehow, after all, people get to write such code?

The code itself was taken from Sedgwick's book on algorithms , began to take a course on coursera.

Answer 1

Do you understand the word degree ?
After all, it's just the n-th number of times the number is multiplied by itself. And the first part to the plus reproduces this. The second part after the plus is and is that plus from the formula.

Answer 2

Such algorithms are easily analyzed and unraveled when you start to manually substitute values. 0, 1, 2,...