Mr. Reader has six different Spiderman comic books, five different Archie comic books and four different Garfield comic books. When stacked, all of the Spiderman comic books are grouped together, all of the Archie comic books are grouped together and all of the Garfield comic books are grouped together. In how many different orders can these 15 comic books be stacked in a pile with the covers facing up and all of them facing the same direction? Express your answer as a whole number.
the three kinds of comics can be arranged in 3! ways
Among each group of n comics of the same type, there are n! ways to arrange them.
So, there are 3!6!5!4! ways to arrange the 15 comics.
To solve this problem, we can first calculate the number of ways to arrange each group of comic books separately, and then multiply the results together to get the total number of arrangements.
Let's start with the Spiderman comic books. Since there are six different books, we can arrange them in 6! (factorial) ways, which is 6 x 5 x 4 x 3 x 2 x 1 = 720.
Similarly, the Archie comic books can be arranged in 5! = 5 x 4 x 3 x 2 x 1 = 120 different ways.
And the Garfield comic books can be arranged in 4! = 4 x 3 x 2 x 1 = 24 different ways.
Now, to calculate the total number of arrangements, we multiply the number of arrangements for each group: 720 x 120 x 24 = 20,736,000.
Therefore, there are 20,736,000 different orders in which these 15 comic books can be stacked in a pile with the covers facing up and all of them facing the same direction.