Hello FatherChaos Originally Posted by

**FatherChaos** Thanks for your thorough answer, Grandad. I still don't understand why the *order* is important in this problem. I mean, what does it matter if a certain pair is placed before the other or if their placement is reversed? As I understand it, the order does not matter in this kind of problem... I'm confused because, in the first two problems, I did not compensate for the lack of order by multiplying with the number of possible permutations...

You are asking three questions here:

- Why does the order matter?
- How did I get numbers 1) and 2) correct?
- Why did we do 3) and 4) in a different way?

The answers are:

- Why does the order matter?

Imagine that the dice are different colours: red, blue, yellow, green, white. Then when you say that there are $\displaystyle 6^5$ ways in which they could land altogether, you are (correctly) taking into account the fact that the same set of 'cards' could appear many times over, because the same set of cards could appear on different coloured dice. You could have, for example, red showing 1, blue showing 2, yellow 3, green 4 and white 5; and this is a different result from, say, swapping over red and blue so that you get red showing 2, blue 1, yellow 3, green 4 and white 5.

You don't normally have to think about this: you just say: there are 6 ways in which the first die can land, 6 ways in which the second can land, ... and so on. So there are $\displaystyle 6^5$ possible ways in which all five can land in total. The fact that you are using words like first, second, ... takes the order into account automatically.

- How did I get the first two right?

Perhaps you can see this now: you used words like first, second, ... as you worked out the number of possibilities. In other words, you (unknowingly perhaps) arranged the dice in order as you made the choices.

- So, why do 3) and 4) differently?

Because there are repeated items, it's easier to choose first, and then arrange, that's all. You can't expect to solve every problem in the same way!

Grandad