Imagine if the letters are distinguishable eg A1, A2, ...A5 etc.

Then the solution would be 14 factorial.

But, as you said they are not distinguishable, so eg

A1,A2,A3,A4,A5 is the same as A2, A1, A3, A4, A5 etc etc.

Thus each AAAAA can be formed in 5! ways.

Similar argument for the Bs and Cs.

So you need to divide 14! by (5!x3!x6!) not (5x3x6) as you suggested.