A student has 3 math books, 4 history books, 2 chemistry books, and 1 Latin book. He wants to arrange them on a bookshelf
a. If all the books have different titles, in how many distinct ways can he arrange them?
b. Throughout parts b-e, assume that all the books from a particular topic have the same title (for example, 3 indistinguishable copies of "Calculus" by Carey). In how many distinct ways can he arrange his books?
c. If he groups the identical math books together, and he groups the identical history books together, and he groups the identical chemistry books together, in how many distinct ways can he arrange his books?
d. If he groups the identical history books together (but isn't picky about the other books), in how many distinct ways can he arrange his books?
e. Given that the identical history books are grouped together, what is the probability that he has grouped each of the identical books (the situation in part c)?