How to explain the division rule in counting problem in a easy way?

The division rule states that "There are n/d ways to do a task if it can be done using a procedure that can be carried out in n ways, and for every way w, exactly d of the n ways correspond to way w"

I really can't understand this definition. Is there a easy way to explain this rule, not using math terms?

Solution 1:

I guess the best way to explain this is with an example.

Let's say we want to arrange $(1,1,2,3,4)$ in a row.

For example,

$$(1,1,2,4,3),(1,1,3,2,4)$$ Are possible arrangements.

We first assume that the two $1$'s are distinct, so we have $5!$ ways to arrange:

$$(1_a,1_b,2,3,4)$$

But for each of the $5!$ ways, we have over-counted some cases:

$$ (1,1,2,3,4) \begin{cases} (1_a,1_b,2,3,4)\\ (1_b,1_a,2,3,4)\\ \end{cases} $$

$$ (2,1,1,3,4) \begin{cases} (2,1_a,1_b,3,4)\\ (2,1_b,1_a,3,4)\\ \end{cases} $$

$$\vdots $$

Every $2!$ of our initial $5!$ ways corresponds to one actual arrangement.

That is why to arrange $(1,1,2,3,4)$ in a row, we have:

$$\frac{5!}{2!} \space \text{ways}$$