How many cases for this case?(Balls and boxes) $Q)$ There are $12$ balls and same two boxes. Among the $12$ balls, $6$ balls are same but the rest of the $6$ are different respectively. (I.e. There are $7$ types of the $12$ balls.) How many cases the balls put into the two boxes?
(Here the case of the empty box allowed.)
$(sol)$ Let the balls $a,a,a,a,a,a,1,2,3,4,5,6$
$(1)$ The "$a$" balls : there are $4$ cases because of the below

*

*Each boxes and the balls are indistinguishable respectively. So $ 0+6 = 1+5 = 2+4= 3 +3$
$(2)$ The "$1$ to $6$" balls : $S(6,1) + S(6,2) = 32$ cases (Here the $S(n,k)$ is the $2$nd stirring number)
So my answer is $4\times 32 = 128$
Are my answer and solution right?
 A: There are indeed $4$ ways to distribute the indistinguishable balls.  $3$ of those suffice to distinguish the two boxes, the $(3,3)$ distribution does not.
For each of the $3$ distributions other than $(3,3)$, we have $2^6=64$ ways to distribute the distinct balls.  That gives us $3\times 64=192$ distributions.
For the $(3,3)$ case we do indeed have $2^5=32$ ways to distribute the distinct balls, as we must divide by $2$ to account for the symmetry.
Thus the answer is $192+32=224$.
A: There are different approaches to this problem. I prefer the following one.
First make the boxes distinguishable. Attach a sticks "1" and "2" to these boxes. And calculate number of possibilities in this case.
There are now 7 possibilities to arrange "a" balls: 0 + 6; 1 + 6; ... ; 6 + 0.
And there are 2^6 = 64 possibilities to arrange "1", ... "6" balls.
Total number of possibilities is 7 * 64.
Note, that all the possibilities can be arranged to pairs: for any arrangement of balls by boxes you can move all the balls from box "1" into box "2" and vice versa - you will get a different arrangement. And these cases become the same case if you remove the sticks from boxes.
That means that the number of cases in the original problem is exactly 1/2 of the number of cases when boxes are different.
And the final answer would be 7 * 64 / 2 = 224.
