I have $5$ different physics textbooks and $4$ different chemistry textbooks. In how many ways can I place the $9$ textbooks on a bookshelf, in a row, if there must be a chemistry textbook in the middle, and there must be a physics textbook at each end

Answer: 57600

Explanation:

Given the following;

5 different physics textbooks

4 different chemistry textbooks

There are 9 different positions for arranging the textbooks on a shelf.

If there must be a physics textbook at both ends;

Number of physics textbooks = 5

Both ends = 2 different positions

Mathematically,

5P2 = 5! : (5-2) !

5! : 3!2! = (5*4)

5P2 = 20 ways

If a chemistry textbook must be in the middle;

Number of chemistry textbooks = 4

Middle = 1 position

Mathematically,

4P1 = 4! : (4-1) !

4P1 = 4! : 3!

4P1 = 4 ways

3 positions have been calculated,

The 6 positions left = 6!

6! = 720ways

Therefore,

6! * 5P2 * 4P1 = 720 * 20 * 4 = 57600