Question

Statistics and probbility

collection consisting of 5 engineering books, 4 algebra books, 3 differential books, 5 physics books ,3 Chemistry books, 4 Biology books to be put together on a shelf.
Find the following:

1- The number of ways to arrange books together on a shelf

2 - The number of ways to arrange books so that the books of each course are together.

3– Number of ways to arrange books so that mathematics books are together and science books together.

4. Number of ways to arrange books so that engineering books are together.

Answer 1

There are 5 Engineering Books, 4 Algebra Books, 3 differential Books, 5 Physics Books, 3 Chemistry Books, 4 Biology Books. There are total- (5+4+3+5+3+4)=24 books.

1. The no. of ways to arrange all books together on a shelf is 24! ways.
2. The number of ways to arrange books so that the books of each course are together-
   There are total 6 different subjects.The no. of ways to arranging them subject wise is: 6! ways
   Now again The Engineering Books can arrange themselves in- 5! ways.
   The Algebra Books can arrange themselves in- 4! ways.
   The Differential Books can arrange themselves in- 3! ways.
   The Physics Books can arrange themselves in- 5! ways.
   The Chemistry Books can arrange themselves in- 3! ways.
   The Biology Books can arrange themselves in- 4! ways.
   The number of ways to arrange books so that the books of each course are together=
   6!*5!*4!*3!*5!*3!*4! ways.
3. We consider Mathematics boks(i.e., Algebra Books and Differential Books) as a single unit and the science books(i.e., Physics Books, Chemistry Books and the Biology Books ) as a single unit.
   So now there are total 7 books. 5 Engineering and 1 Mathematics and 1 Science. They can be arranged in 7! ways.
   There are total 7 Mathematics books. They can arraange themselves in 7! ways.
   There are total 12 Science books. They can arraange themselves in 12! ways.
   No. of ways  to arrange books so that mathematics books are together and science books together: 7!*7!*12!.
4. We consider 5 Engineering Books as a single unit.
   So now there are total 20 books. They can be arranged in 20! ways.
   The 5 Engineering Books can be arranged themselves in 5! ways.
   Number of ways to arrange books so that engineering books are together- 20!*5!

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