Suppose there are 4 books on fairy tales, 5 novels and 3 plays. They have to be arranged so that the books on fairy tales are together, novels are together and plays are together, but we no longer require that they should be in a specific order. In how many ways can this be done?
First, we consider the books on fairy tales, novels and plays as single objects.
These three objects can be arranged in 3!=6 ways.
Let us fix one of these 6 arrangements.
This may give us a specific order, say, novels -> fairy tales -> plays.
Given this order, the books on the same subject can be arranged as follows.
The 4 books on fairy tales can be arranged among themselves in 4!=24 ways.
The 5 novels can be arranged in 5!=120 ways.
The 3 plays can be arranged in 3!=6 ways.
For a given order, the books can be arranged in 24*120*6=17280 ways.
Therefore, for all the 6 possible orders the books can be arranged in 6*17280= 103680 ways.
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