Arranging letters with terms This is a question I found in an old test while I was preparing for a test I have in a few days.
The lecturer didn't really show us how to arrange with certain terms, so I need help with solving this.
Question 1 :
How much arrangements are there for the word SOCIOLOGICAL when the letters AG have to be together.
Question 2 :
Same question, only a different term : when all the letters A,O,I have to be together.
Thanks.
 A: A common method for these sorts of problems to arrange the unrestricted characters, then figure out where you're allowed to put the rest.  Think of the spaces between already-arranged letters as bins, in which to put the rest.
So for question 1: arrange the letters SOCIOLOICL in any way; there are 
$$
\binom{10}{1,3,2,2,2}=\frac{10!}{1!\cdot 3!\cdot 2!\cdot 2!\cdot 2!}
$$
ways to do this, since the multiplicities are 1(S), 3(O), 2(C), 2(I), and 2(L). Now, where can you put the A and the G?  Well, of the 11 bins (9 between letters, 1 before the whole word, and 1 after the whole word), you must put them in the same one; and they can be arranged either AG or GA. So, there are $11\cdot2=22$ ways to position A and G, for a total of
$$
\binom{10}{1,3,2,2,2}\cdot22
$$
ways.
For question 2: if both I's must be together and all three O's must be together, we can think of it this way: reduce our letters so that there is only one I and only one O.  Arrange the resulting set of letters. Then expand the I to be II and the O to be OOO. 
Reducing the letters yields SOCILGCAL; these can be arranged in
$$
\binom{9}{1,1,2,1,2,1,1}
$$
ways.  Then we expand the I and O, which can only be done in one way.  So, this is the final answer here.
