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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?

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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!

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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