# How many ways can 1 arrange on bookshelf 5 thick books,4 medium\par sized books and 3 thin books so that the books of the same size remain together?

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We have 5 think books , 4 medium books and 3 thin books to be arranged of a shelf.

The books of the same thickness must be arranged together.

To arrange the thick books we have 5*5 ways = 25 way

to arrange the medium books we have 4*4 ways = 16

To arrange the thin books we have 3*3 = 9 ways

Now the number of ways are:

25 + 16 + 9 = 50 ways

Since we have 3 groups of books( thick , medium, and thin), then there are (2*3=6) ways to arrange on the shelf

**==> The number of ways are 6*50 = 300 ways**

There are 3 groups.

The 3 groups could permuted in 3! ways or 1*2*3 = 6 ways.

Within the first group of 5 thick books we can permute the books in5! ways.

Within the medium group of 4 books the number of possible arrangements is 4!.

Within the group of 3 thin books the number of possible arrangements is 3!.

Therefore , the number of arrangements under the given condtion is 3!*5!*4!*3! = (1*2*3)(1*2*3*4*5)(1*2*3*4)(1*2*3) = 6*120*24*6 = 103680 ways.