So when I say S4=4, what I'm saying is that every single unique sequence where we have four matching pairs will be counted 4 times each.
When we do 4C1*7!/2!2!2!, it's counting all sequences where aa is together, all sequences where bb is together, all sequences where cc is together, and all sequences where dd is together.
If we examine a specific sequence, like aabbccdd,
It will get counted when we count how many sequences have aa, it will also get counted when we count how many sequences have bb, cc, and dd. So we're counting this sequence 4 times, but it only should count as one.
But this is true of EVERY sequence like that. bbaaccdd also gets counted four times because it also is counted each time we count the aa's bb's cc's and dd's.
Picture a 4-circle venn diagram. The region that contains all four circles is the set of sequences that have four adjacent matching pairs. So when we add all the ones that have aa, all the ones that have bb, all the ones that have cc, all the ones that have dd, that region is getting counted 4 times, and that is why S4=4.
I'm not 100% sure I understood your question, so I hope this helps.
You are right that aabbccdd is different from bbaaccdd. My point is just that each of these unique sequences is counted four times when we do 4C1*7!/2!2!2!
When we do 4C1*7!/2!2!2!, it's counting all sequences where aa is together, all sequences where bb is together, all sequences where cc is together, and all sequences where dd is together.
If we examine a specific sequence, like aabbccdd,
It will get counted when we count how many sequences have aa, it will also get counted when we count how many sequences have bb, cc, and dd. So we're counting this sequence 4 times, but it only should count as one.
But this is true of EVERY sequence like that. bbaaccdd also gets counted four times because it also is counted each time we count the aa's bb's cc's and dd's.
Picture a 4-circle venn diagram. The region that contains all four circles is the set of sequences that have four adjacent matching pairs. So when we add all the ones that have aa, all the ones that have bb, all the ones that have cc, all the ones that have dd, that region is getting counted 4 times, and that is why S4=4.
I'm not 100% sure I understood your question, so I hope this helps.
You are right that aabbccdd is different from bbaaccdd. My point is just that each of these unique sequences is counted four times when we do 4C1*7!/2!2!2!
