I am interested in arrangements 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

I define the following sets

$\displaystyle A_1$ = {arrangements with both T's before both A's}

$\displaystyle A_2$ = {arrangements with both A's before both M's}

$\displaystyle A_3$ = {arrangements with both M's before E}

I am able to find the cardinality of everything else but $\displaystyle A_1 \cap A_3$

i.e. arrangements with both T's before both A's and both M's before E.

I know I want to pick four of the 11 spots (11 choose 4)

and arrange H,I,C,S in those spots (4! ways)

But I dont know whaat to do next, I can't place only the T's in the two left most spots because certainly go T,M,M,E,T.

Thank you for your help.