Mathematical Concept In Multi-Dimensional Array Sorting

I have found this code written online(not by me), and was hoping to get an answer on what the mathematical formula or concept is that makes this function work. I am curious as to how this person designed this. First, I will explain the requirements that the function must produce, then I will supply the code, and a link to a working code pen for further hacking. P.S. The problem uses the word "vector", but since this is Javascript, vector just means array.

Function Requirements

Given a vector of vectors of words, ex. [['quick', 'lazy'], ['brown', 'black', 'grey'], ['fox', 'dog']].

Write a function that prints all combinations of one word from the first vector, one word from the second vector, etc. The solution may not use recursion. The number of vectors and number of elements within each vector may vary.

Example output: 'quick, brown, dog', 'lazy black fox' etc.

My Current Level Of Understanding I am already aware that by using the principle of multiplication, to find the number of possible combinations available in this scenario is to just multiply the lengths of each inner vector by each other. For this specific example, we get a total of 12(2x3x2) different possible combinations. Where I fall off however, is inside the 4 nested for loops section of the program.

Whoever wrote the code, clearly understands some concept or formula that I do not. Just two examples, are the "previous" variable used inside the loops, and the strategic placement of where they decide to increment the j variable. It seems to me that they might be aware of some mathematical formula.

Code Below is the code without comments. If you however go to this codepen, I have included the same code with plenty of comments that explain how the program works, so you don't have to trace everything out from scratch. You can also test the output in the built-in console.

function comboMaker(vector) {

  var length = vector.length;

  var solutions = 1;
  for (var i = 0; i < length; i++) {
    solutions *= vector[i].length;
  }

  var combinations = [];
  for (var i = 0; i < solutions; i++) {
    combinations[i] = [];
  }

  var previous = 1;

  for (var i = 0; i < length; i++) {

    for (var j = 0; j < solutions;) {

      var wordCount = vector[i].length;
      previous *= vector[i].length;

      for (var l = 0; l < wordCount; l++) {

        for (var k = 0; k < (solutions/previous); k++) {
          combinations[j][i] = vector[i][l];
          j++
        }
      }
    }
  }

  for (var i = 0; i < solutions; i++) {
    console.log(combinations[i].join(" "));
  }
}

comboMaker([['quick', 'lazy'], ['brown', 'black', 'grey'], ['fox', 'dog']]);