Multiplication Rule

Question

Multiplication Rule for Counting

Why does "j" have to be smaller or equal to "k"? Doesn't it work regardless?

Answer 1

It looks like $j$ is just indexing the $k$ choices.

You have $k$ choices total, and they're using the $j$'s as a way to refer to some ordering of the choices so that they can define what $n_1 n_2 \cdots n_k$ means.

Answer 2

j stands for the position of the number, while k is the total number. So for instance we have 5 numbers , 1, 3, 4, 6, 10. Then k is 5. So the number 1 will be $m_1$, since it is the first position in the set of numbers. So j here is one for the number 1. Evidently, j(position of number) will not be greater than k(number of numbers).