I have 5 different biology textbooks and 4 different economics textbooks. In how many ways can I place the 9 textbooks on a bookshelf, in a row, if there must be an economics textbook in the middle, and there must be a biology textbook at each end?
1440 and 172800 are wrong
We have 9 "slots" or blanks we want to fill these 9 textbooks into. Here's a visual:
_ _ _ _ _ _ _ _ _ and each one can be thought of as holding one of these textbooks; be it biology or economics.
First, we want to consider ou restrictions; the problem says that there must be an economics textbook in the middle and a biology textbook at each end. We look to our first requirement, which is the economics textbook in the middle. It makes sense for us to fill in this spot first. We then have:
_ _ _ _ 4 _ _ _ _ ways for us to fill in the middle spot because there are 4 economics textbooks to choose from. Our next requirement or restriction is that there must be a biology textbook at each end. Let's then fill in these.
5 _ _ _ 4 _ _ _ 4, ways to do so, since in the first spot, there are 5 biology textbooks(out of 5) to choose from, and in the last slot, there are 4 left over(since you've picked one already).
After this, we have no more restrictions left. At this point, we have already placed 3 books, so that leaves us with 6 remaining. We can then fill out these slots however we want, which gives us:
5 6 5 4 3 2 1 4 number of ways to do these arrangements. Next, we just multiply all of these, which gets us:
5 * 6 * 5 * 4 * 3 * 2 * 1 * 4 = 14400 ways