Question

I have 7 different calculus textbooks and 4 different number theory text books. In how many ways can I place the 11 text books on a bookshelf, in a row, if there must be a number theory text book in the middle, and there must be a calculus textbook at each end.

Answer 1

There are 7 ways to choose the calculus textbook at one end, then 6 ways to choose the calculus textbook at the other end.  Then there are 9 ways to choose the second textbook, 8 ways to choose the third textbook, and so on.  But then there are 4 ways to choose the number theory textbook, so there are 7*6*9!*4 = 60963840 ways that work.