Mai has moved into a new house and now she is organising her library. She wants to put 8 books on one shelf (each book has unique name), also she has a very picky taste. She has red, blue, green and yellow books, and from each color there are two books . She dislikes when red book is next to a green one, and when blue book is next to a yellow one. Also, she would like to avoid same-coloured books being neighbouring . On how many ways can Mai organise her library?
Important: Combination ABCDEFGH and HGFEDCBA will be considered as one combination.
To sum up:
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Solution:
Basically, after every red/green book is a blue/yellow book and vice versa. Therefore, we can divide books into two groups, the red/green and blue/yellow group. So, on the "place" for 1st book, we can put any book (8, because we do not have a book before the first one, so nothing oblige us). On 2nd place, we must put book from the other group, different from the 1st book's group (4). 3rd book will be from the 1st book's group. 4th book is same group as 2nd, and so on....
At the end, we will have multiplication: 8 × 4 ! × 3 ! . But, we have to be careful and to consider note that is marked important , because of it, the multiplication must be divided by 2 (avoiding symmetry). Then, solution is: ( 4 ! ) 2 = 2 4 2
That is 5 7 6 .