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Question

A total of 5 different mathematics books, 4 different physics books and 2 different chemistry books are to be arranged in a row in a book shelf. Which of the following is (are) TRUE?
  1. The number of arrangements in which two chemistry books are separated is 9×10!
  2. The number of arrangements in which four physics books are together is 8! 4!
  3. The number of arrangements in which no two mathematics books are together is (76)(6!)
  4. The number of arrangements in which the books of the same subject are all together is 12(4!5!)


Solution

The correct options are
A The number of arrangements in which two chemistry books are separated is 9×10!
B The number of arrangements in which four physics books are together is 8! 4!
D The number of arrangements in which the books of the same subject are all together is 12(4!5!)
Total number of arrangements =11!
Number of arrangements that two chemistry books are together =2!×10!
Hence, number of arrangements that two chemistry books are separated =11!2!×10!=9×10!

Number of arrangements in which four physics books are together =8! 4! since four physics books can also be permuted among themselves in 4!

Similarly, number of arrangements in which the books of the same subject are all together =3!(5! 4! 2!)=12(4!5!)

We want no two mathematics books should be together.
So, let us first place the other six books.
__  __  __  __  __  __
This can be done in 6! ways as there are six blanks.
Now, we can put mathematics books between these 7 gaps.
This can be done in 7C5×5!
Hence, number of arrangements in which no two mathematics books are together is 6!×7C5×5!

 

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