Elementary counting: multiplication?

Question

It is certainly a silly question.

When we store $n$ students in $N$ different day, we say the following for the first student there is $N$ different choices, for the second $N-1$, etc. So that there is $$N\cdot (N-1)\cdots(N-n+1)$$ possibilities.

But I never really understanding why we do a multiplication.

For exemple let's say we have $2$ students Bob and Lea and $3$ days. For Bob there is $3$ choices so that for Lea there is $2$ choices. And then $3+2=5$ choices and the "correct" answer is $3\times 2=6.$

I cannot think where I am wrong.

Answer 1

You would add the numbers of choices in case they exclude each other. E.g. if you could choose between Bob on one of three days, or Lea on one of two days.

$$B1\text{ or }B2\text{ or }B3\text{ or }L1\text{ or }L2.$$

This is not the case here and the choices for Bob and Lea can be combined as

$$B1L2\text{ or }B1L3\text{ or }B2L1\text{ or }B2L3\text{ or }B3L1\text{ or }B3L2.$$

You can display that in a tabular form

$$\begin{matrix}&B1&B2&B3\\L1&&\circ&\circ\\L2&\circ&&\circ\\L3&\circ&\circ&\end{matrix}$$ where there are three rows and two dots per row.