How many distinct ways are there to arrange the letters in the word parallel so that:
(a) The "P" and the "R" occur next to each other;
(b) The "P" and the "R" do not occur adjacent to each other;
(c) There are no consecutive L's
This type of question boils down to how many ways can you arrange things, some of which are indistinguishable from each other. (i.e., the l's and a's in parallel can't be told apart, so the number of ways to arrange the letters isn't simply 8!).
In general, the arrangement of n items, of which n1 are alike, n2 are alike, ..., ni are alike ...
The solution discusses permutations and provides steps necessary to find the maximum number of distinct arrangements.