meaning of some words used in Theorem of Counting Rule for Compound Events

Question

I am reading statistics. There is a theorem in my textbook.

The theorem about Counting Rule for Compound Events is as the following...

If an operation consists of k steps, of which the first can be done in n1 ways,

for each of these the second step can be done in n2 ways,

for each of the first two the third step can be done in n3 ways,

and so forth,

then the whole operation can be done in n1*n2*...*nk ways.

I have 2 questions

What does "each of these" mean?

 1. "each way of these n1 ways"
 2. "each one of these n1 ways"

What does "each of the first two" mean?

 1. "each way of the first two ways"
 2. "each one of the first two ways"
 3. "each way of the first two steps"
 4. "each one of the first two steps"

Thank you for reading.

I guess the answer to question 2 might be

"for each (way of the n1∙n2 ways) of the first two (steps)"

Just a note to remind myself in the future.

Answer 1

I won’t directly answer your question, because no concise wording is necessarily perfectly clear.

Perhaps a clearer statement of the requirements would be

If ... regardless of which way the first step is done, the second step can be done in $n_2$ different ways, and regardless of how the first two steps are done, the third step can be done in $n_3$ different ways, ...

The point is that this rule applies if the number of different ways in which the $i$-th step can be done is independent of the particular choices made for the preceding steps.

An example where this applies is if you were creating license plates of the form digit-letter-digit (10 digits to choose from and 26 letters) with no restrictions, as a three-step process of choosing the first digit, then the letter, then the final digit. Regardless of which digit is chosen as the first digit, there are 26 choices for the letter, and regardless of how the first two steps are done, there are 10 choices for the final digit.

On the other hand, if you were creating all license plates of the form digit-letter-digit where the letter O (“oh”) could not be adjacent to the digit 0 (zero), the number of choices (of letter) at the second step would not always be 26. It would be 25 if the particular choice of 0 (zero) were made as the first step, and 26 otherwise, so this counting principal would not apply. (In addition, the number of ways in which the final digit can be chosen is 9 if the letter O (oh) is chosen at step two, but 10 otherwise.)