How many arrangements are there of MATHEMATICS with both T's before both A's or both A's before both M's or both M's before the E ?
Can someone also point to some online resource that has such practice questions ?
Thanks.
Edit:
I have some initial thought, please tell me if I am in the right direction :
For the case both T's before both A's :
We can picture the permutation as follows :
(7) T (7) T (7) A (7) A (7) (Remaining letters: M H E M I C S – total 7)
Each of the brackets are empty spaces for now. We need to pick 7 out of the total 7 X 5 = 35 spaces; that will ensure both T's being before both A's.
So number of ways = C(35,7) X 7!/2! (because two M's are same – we need to divide by 2!)
Best Answer
You were headed in the right direction, but you went way too far. The $C(35,7)$ is much too much. It counts, for example, "-M---I-T..." and "--M-I--T..." as different, even though both are basically "MIT..."
Here's a way to think of it. Starting from TTAA, allow yourself to insert the other letters, C, E, H, I, M, M, and S, one at a time, anywhere before, between, or after letters already in position. (It might help to picture the second M as, temporarily, an N, remembering to divide by $2$ when you're done.) The C have $5$ places it can go, the E then has $6$, the H has $7$, and so forth, for a total of
$$(5\cdot6\cdot7\cdot8\cdot9\cdot10\cdot11)/2$$
Can you now do the other examples you asked about (both A's before both M's or both M's before the E)? The first of those should actually be very easy!