Can anyone help me with this task?
We throw with $6$ classic (cubical) dice. Two are yellow, two are blue and two are green. After every throw, we put them into a line. How many different lines exist if we distinguish colors and the values?
I think that answer is $$1440\text{ (options for yellow)}\times1440\text{ (options for blue)}\times1440\text{ (options for green).}$$ But sadly, I don't think that this is correct.
Thank you.
First, independent of the colours, we can have $6^6$ arrangements of eyes. (Number of values a die can take to the power of the number of dice.) In addition to this, you need to consider the arrangement of colours. If you first distribute two dice (e.g. yellow) to the $6$ slots, you get $(5+4+3+2+1)$ arrangements. Then, when you distribute the next pair (e.g. blue) to the $4$ remaining slots, there are $(3+2+1)$ ways. And the final pair has to fit in the remaining two slots.
In total that makes
$$6^6(5+4+3+2+1)(3+2+1) = 4199040$$
possible arrangements.
Is this what you were looking for?