# Elementary counting: multiplication?

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.

• The choices are independent of each other. Independence means multiplication. Jun 7, 2017 at 9:43

$$B1\text{ or }B2\text{ or }B3\text{ or }L1\text{ or }L2.$$
$$B1L2\text{ or }B1L3\text{ or }B2L1\text{ or }B2L3\text{ or }B3L1\text{ or }B3L2.$$
$$\begin{matrix}&B1&B2&B3\\L1&&\circ&\circ\\L2&\circ&&\circ\\L3&\circ&\circ&\end{matrix}$$ where there are three rows and two dots per row.