So I am confused over why the sample space of rolling a red die and green die results in (1,4) being different from (4,1), but there can only be one (1,1). Why can't there be (1 -red, 1-green) and (1-green, 1-red), if order matters?
In addition, does the generalized counting principle always account for all outcomes possible? I am not sure if the generalized counting principle is counting total number of outcomes where order matters, or order does not matter. It seems that it changes depending on the problem, so I am confused on how to properly apply the generalized counting principle.
For example, my textbook says: Let E 1 , E 2 , . . . , E k be sets with n 1 , n 2 , . . . , n k elements, respectively. Then there are n 1 × n 2 × n 3 × · · · × n k ways in which we can, first, choose an element of E 1 , then an element of E 2 , then an element of E 3 , . . . , and finally an element of E k .
So it seems to me that order matters for the generalized counting principle? Why don't we multiply by k!, because there are also k! ways to order these Ek sets?
Thank you in advance!