Discrete Math problem combinations and restrictions I am trying to get my head around an idea but I can't seem to get it to work.
Imagine you have the word "MAMMAL"
Lets see I wanted to figure out how many ways I could rearrange the letters. Well that is easy. It is simply
6!/3!2! = 60 possibilities
But what if I added a restriction to it such that all M's must be together.
Then I could consider all M's as one letter and it would be 4!/2! = 12 possibilities.
Again I understand that. But what if the restriction was there needs to be a minimum of 2 M's together at all times.
If I were to do the same process as the previous example, I would combine 2 M's into one. Which then would become 5!/2! = 60.
This seems to be a wrong answer because it is the same as my first calculation of finding all possibilities without any restrictions. Can anyone please explain to me as to how I need to approach the last problem of finding number of combinations where at least 2 M's are always together? Thanks!
 A: Consider the complimentary event, which is no Ms are together, i.e. all Ms are separate.
3 Ms need to be put in 6 places. There are only 4 choices, M _ M _ M _,  _ M _ M _M, M _ M _ _ M and M _ _ M _ M.
In remaining 3 places, you need to put 3 letters, out of which 2 are different. Therefore, there are $4\times\dfrac{3!}{2!} = 12$ such words ($4$ for $4$ different arrangements described above).
Subtract this from total words $60$, and you'll get your answer as $48$.
A: The reason your answer is incorrect is the following: When you consider MM one letter, then you need to remember there is another M around too. So for example, you counted MMMAAL and MMMAAL separately which is why you got $60$, more than what you should've gotten. 
In order to rectify that, just begin with $5!/2!$ and then subtract the doubly counted ones. Notice that for all the cases where there are three M's together, you get this double counting situation. There are 12 cases where the three M's are together so you counted 12 of them twice therefore you subtract 12 to get $60 - 12 =48$ as your answer.
