Suppose we have 7 slots to fill with the letters A, B and C. We can pick A two times, B three times and C two times. How many combinations can be made?
My thought was to group A with C, to essentially find how to fill 4 out of the 7 slots, so we get $C_4(7) = {7 \choose 4} = 35$ and call it a day, but it can't be that easy.
What am i missing?