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?
Let's start by looking at the "restrictions" imposed within the problem. First is that there must be a chemistry textbook in the middle. At this stage of the problem, let's envision the 9 textbooks as 9 blanks, with the middle one having to be chemistry:
_ _ _ _ C _ _ _ _
Since we know it must be a chemistry book, that gives us 4 options for how many chemistry books could go into that particular "blank".
Next, let's look at the next restriction, which is that there must be a physics textbook at each end. The implication of this is that the blanks would look something like this:
P _ _ _ 4 _ _ _ P
With P being the physics textbooks. Since we know there are 5 physics textbooks total, the possibilites for the first physics "blank" are 5 books to choose from, with the next one being 4 to choose from. At this stage of the problem, we have :
5 _ _ _ 4 _ _ _ 4
, with the numbers representing how many ways there are to satisfy each respective restriction. After these 2 restrictions, we are left with 6 books(9-3) which we are free to place however. As such, the rest of the problem is just simple multiplication, with the final result looking like this:
6 6 5 4 4 3 2 1 4. Multiple these, and you get 6*6*5*4*4*3*2*1*4 = 69120