Total number of possible arrangements without keeping any row devoid of elements I am given with 6 symbols- +, +, x, x, , . How many possible arrangements of these symbols into the below figure is possible if no row should be empty? ![Figure in which symbols are to be inserted]( I have tried to proceed with this in several ways like finding total number of possible arrangements and then subtracting by those not in the same rows. But it won't work. Also, it is pretty clear that the boxes in second and fourth row have to be filled at all events. So, we are left to place one element in 6 boxes. I choose and permute but at the end the answer still doesn't come right.

$\underline{Method\;\; avoiding\;\; casework:}$

Taking the symbols to be $AABBCD$ and straightening the $\Bbb{E}$ pattern for simplicity,
permute the symbols and put them in $5$ boxes, e.g. one permutation could read $\boxed{AB-}\boxed{D}\boxed{B--}\boxed{C}\boxed{-A-}$

There are $\frac{6!}{2!2!} = 180$ permutations of symbols,
$3\binom32 = 9$ ways of choosing box for two symbols, and placing them,
and $3\cdot3 = 9$ ways for placing a symbol in the other two "triple" boxes

Putting it all together, we get a total of $180\cdot81$ ways