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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?

A
The number of arrangements in which two chemistry books are separated is 9×10!
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B
The number of arrangements in which four physics books are together is 8! 4!
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C
The number of arrangements in which no two mathematics books are together is (76)(6!)
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D
The number of arrangements in which the books of the same subject are all together is 12(4!5!)
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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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