Find the number ofpermutations of the given letters "AAABBCC''.distinguishable

First I thought I could just do this :

A_{1}A_{2}A_{3}B_{1}B_{2}C_{1}C_{2 }

""""""""""""" C_{2}C_{1}

"""""""""B_{2}B_{1}C_{1}C_{2}

""""""""""""" C_{2}C_{1}

A_{1}A_{3}A_{2}B_{1}B_{2}C_{1}C_{2}...

so 2 start with A_{1}, 2 with A_{2}, and 2 with A_{2 }

then for every set of A there are two sets of B

and for every set of B there are two more sets of C

But apparently it's wrong.

Can anyone tell me why and more importantly where does this formula 7!/(3!2!2!) come from.